What's new in higher dimensions? This is a very speculative/soft question; please keep this in mind when reading it. Here "higher" means "greater than 3".
What I am wondering about is what new geometrical phenomena are there in higher dimensions. When I say new I mean phenomena which are counterintuitive or not analogous to their lower dimensional counterparts. A good example could be hypersphere packing.
My main (and sad) impression is that almost all phenomena in higher dimensions could be thought intuitively by dimensional analogy. See for example, this link:

What this implies (for me) is the boring consequence that there is no new conceptual richness in higher dimensional geometry beyond the fact than the  numbers are larger (for example my field of study is string compactifications and though, at first sight, it could sound spectacular to use orientifolding which set a loci of fixed points which are O3 and O7 planes; the reasoning is pretty much the same as in lower dimensions...)
However the question of higher dimensional geometry is very related (for me) to the idea of beauty and complexity: these projections to 2-D of higher dimensional objects totally amazes me (for example this orthonormal projection of a 12-cube) and makes me think there must be interesting higher dimensional phenomena...

I would thank anyone who could give me examples of beautiful ideas implying “visualization” of higher dimensional geometry…
 A: In dimensions $d > 4$ there are just the obvious three regular polytopes: the simplex, the hypercube and its dual, the cross polytope. The fourth dimension stars three more, the 24-cell, the 120-cell and the 600-cell. 
Pack spheres of diameter $1/2$ in the corners of a unit hypercube in dimension $d$. Then inscribe a sphere $S$ in the center of that cube tangent to the corner spheres. The long diagonal of the hypercube has length $\sqrt{d}$. It follows that the diameter of $S$ is $(\sqrt{d} -1)/2$. When $d=9$, $S$ is tangent to the facets of the hypercube. When $d> 9$  it sticks out past the facets.
 
$4x + 1 = \sqrt{d}$ so $2x = \frac{\sqrt{d} -1}{2}$.
My thesis advisor Andy Gleason told me once that he'd "give a lot for one good look at the fourth dimension".
A: There is already a great gap between 2 and 3 dimensions.
Here is a mildly interesting but very simple one I came up with, when thinking about this very question some time ago:

There is no partition of the 2d-plane into a finite number of path-connected regions that are translations of one another. But there is a partition of the 3d-space into just two path-connected regions that are translations of one another!

Another one is:

The mean value theorem holds for 1 and 2 dimensions, but not in 3, because of a spiral.

A: Exotic spheres are a feature only of dimensions higher than 3.  These are topological spaces which are homeomorphic to a sphere, but with different differential structure.  Informally, you can paraphrase this result as "there are multiple distinct ways to do calculus on high dimensional spheres, but only one way on low dimensional spheres."
A related concept is exotic $\mathbb{R}^4$, but this is not characteristic of other high dimensional spaces--only $\mathbb{R}^4$.

Concentration of measure is a phenomenon that is common to many high dimensional geometric objects.  The basic idea here is that most of the mass of  many typical high dimensional objects (for example, the sphere) is concentrated near relatively small subsets.  For instance, on the sphere, most of the mass is near the equator (or any other high dimensional analog of a "great circle").

As perhaps the best example of a phenomenon that "looks" very different in high dimensions, I suggest knot theory.  The first hint that there are surprises in store is the observation that knots in the usual sense (embeddings of $S^1$ whose complement has non-trivial topology) do not exist in dimensions other than 3.  However, there are analogous objects in higher dimensions: A $k$ sphere can be embedded in a $k+2$ sphere to form a sort of "high dimensional knot".  So the important thing is that the "knot" has codimension 2 in the ambient space.  Try thinking about what these high dimensional knots look like :)

If you are willing to consider infinite dimensions, then all bets are off. 
 Finite intuition can be downright dangerous in this case.  For example: 


*

*For finite dimensional vector spaces, all norms are equivalent, but this is not true in infinite dimensions.  

*There is no translation invariant measure in infinite dimensions.  This is due to the geometric fact that you can put infinitely many balls of radius $1/2$ inside a ball of radius $2$ in infinite dimensions.

A: It is tempting to think that higher dimensions only bring "new geometrical phenomena", but they can also take away familiar properties often considered as elementary in lower dimensions.
Something we take for granted in 3D, for example, is the vector cross-product as a vector-valued, bilinear, anti-commutative, product of two 3D vectors. It may come as a surprise that there is no direct equivalent of the cross-product in $\,\mathbb{R}^n\,$ for $\,n>3\,$, except for $\,n=7\,$.
That "the seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions" is not immediately obvious, and the reason why the familiar cross-product only exists in $\,3\,$ or $\,7\,$ dimensions goes to the deeper Hurwitz' theorem which implies the only finite-dimensional normed division algebras must have  dimension $1$, $2$, $4$, or $8$.

[ EDIT ]  Further insights can be gleaned from the related question Cross product in $\mathbb R^n$, some answers under Is the vector cross product only defined for 3D?, the quora question Why does the cross product exist only in three and seven dimensions? and J.M. Massey's article Cross Products of Vectors in Higher Dimensional Euclidean Spaces in The American Mathematical Monthly
Vol. 90, No. 10 (Dec., 1983), pp. 697-701.
A: A sort of trivial thing that I haven't seen mentioned in any of the other answers yet is that rotations get weird when $d \ge 4$, messing with our three-dimensional intuition and requiring the development of completely new intuitions.
Specifically, in our three-dimensional world, we're used to things rotating around an invariant axis.  The fact that in two dimensions things rotate around a point instead just seems like a special case of this — the invariant axis is still there, if we imagine the 2D space as being embedded in normal 3D space; it's just pointing out of the plane.
But in four dimensions, things don't rotate around an invariant axis; they rotate around an invariant plane.  And what's worse, that invariant plane can accommodate another independent rotation at the same time!  So things in four dimensions can rotate at two different speeds in two different, orthogonal planes at the same time, with only the single intersection point of the two planes staying in place.
This makes imagining elementary physics in four dimensions pretty weird.
For example, if the Earth was four-dimensional, it wouldn't have two poles and an equator; instead, it might have two equators, both orthogonal and equidistant to each other.  And the 4D Moon's orbit around the 4D Earth (never mind the stability of orbits in four dimensions in the first place) could be at a constant 90° angle to the Earth's orbit around the Sun.  And I won't even try to imagine what a 4D galaxy would look like, although it definitely wouldn't be a flattened disc like our own Milky Way.
Also, people living on the 4D Earth might have a hard time inventing the wheel.  Certainly their wheels would look nothing like ours, what with having no axle (or, alternatively, perhaps having two orthogonal axles, or even an entire flat plane for an axle).
In higher dimensions, things get a bit more complicated yet.  A general 5D rotation has two independent planes of rotation and an invariant axis, while six dimensions are enough to accommodate three independent planes of rotation.  More generally, $2n$ dimensions are just enough to let an object rotate in $n$ orthogonal planes at the same time, while in $2n+1$ dimensions there will always be an extra invariant axis left over.
A: In addition to the good answers you have got so far, I think the tiling problem also fits in the scope of this question. In higher dimensions (and even in the 3D space), the tiling problem becomes highly complex and some strange results appear.
One seemingly obvious observation that becomes invalid in higher dimensions is Keller's conjecture, which states that

In any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face. For instance, as shown in figure below, in any tiling of the plane by identical squares, some two squares must meet edge to edge.


And I quote the rest from wikipedia:

It was shown to be true in dimensions at most 6 by Perron (1940). However, for higher dimensions it is false, as was shown in dimensions at least 10 by Lagarias and Shor (1992) and in dimensions at least 8 by Mackey (2002), using a reformulation of the problem in terms of the clique number of certain graphs now known as Keller graphs. Although this graph-theoretic version of the conjecture is now resolved for all dimensions, Keller's original cube-tiling conjecture remains open in dimension 7.

More precisely, the reduction to Keller graphs is equivalent to making the assumption that all hypercubes have integer or half-integer coordinates. (For example, this may be true in the image above if we assume that each vertical stripe is shifted exactly half a square's height from the stripe to its left.) So the progress on the problem to date can be summarized as:


*

*Lagarias and Shor showed that even under this restriction, counterexamples to Keller's conjecture can be found in $d \ge 10$ dimensions.

*Mackey extended this result to $d = 8$ dimensions (which implies $d=9$, too, since we can stack a bunch of 8-dimensional tilings on top of each other, shifted so none of the hypercubes in adjacent layers share faces).

*However, Debroni et al. (2010) showed that all tilings with half-integer coordinates  in $d=7$ do satisfy this conjecture.


This leaves the $d=7$ case open, but implies that if a counterexample does exist in $7$ dimensions, it must look weird and have less structure than any of the higher-dimensional counterexamples we've found.
A: The following is very low-level difference, but a surprising one nonetheless.
A plane is a $2$-dimensional affine subspace (like the $xy$-plane) of $\mathbb{R}^n$. In three dimensions, it is obvious that two planes either are parallel, either intersect in a line or are exactly the same. It is impossible for two planes to intersect in only one point.
Going to $\mathbb{R}^4$ this fails. Indeed, the $xy$-plane and $zt$-plane only intersect in the origin. Any visual intuition you have becomes useless in higher dimensions as we can only visualize $3$-dimensional phenomena. On the other hand, using coordinates (as I did) it is trivial to understand that two planes in higher dimensional spaces can intersect in only one point.
I almost feel bad for posting such an obvious thing, but this difference can't be visually intuitive. 
A: The fundamental group of (connected) $n$-manifolds has a change around $n=3$.  For $n=2$, the fundamental groups are cartesian products of various combinations of $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}$.  For $n=3$, these are automatic groups.  For $n=4$, any finitely presented group can be the fundamental group.  So, determining triviality of the fundamental group, for the three values of $n$ mentioned, is trivial, decidable, and undecidable, respectively.  If we let the dimension go to infinity, we can't even promise to have a finite presentation.
Triangulations also change with dimension.  Differentiable manifolds (any dimension) have piecewise-linear (PL) triangulations.  In 2 and 3 dimensions, any topological manifold has a triangulation, and the triangulations of a given manifold are piecewise-linear equivalent.  In fact, in 2 and 3 dimensions, each manifold has a smooth structure, unique up to diffeomorphism.
(For (complete, finite volume) hyperbolic manifolds of any dimension $\geq 3$, this is the (generalized) Mostow rigidity theorem -- every isotopy is an isometry.)  In dimension 4, the $E_8$ manifold has no PL triangulation and many manifolds have infinitely many PL-inequivalent triangulations.  In dimensions $\geq 4$, only some manifolds have PL triangulations (ibid.).  In dimensions $\geq 5$, only some manifolds are homeomorphic to a simplicial complex -- most have no form of triangulation.  (This paragraph summarizes the last paragraph of Triangulation (topology): Piecewise linear structures in the English language Wikipedia.)
And, one that surprised me the first time I ran across it in an application ($n$-dimensional diffusion, many years ago), the distribution of angles between random (nonzero) vectors in $n$-dimensional Euclidean space is concentrated (and becomes more concentrated with higher $n$) around $\pi/2$ -- everything is nearly orthogonal to everything else.  This has an analog in $n=3$ -- the spherical zone between latitudes $+\pi/4$ and $-\pi/4$ has more area than the sum of the spherical caps above and below the zone.  This becomes rapidly moreso as the dimension increases: the $\pm$ latitude where the measure of the zone equals the measure of the two caps goes to zero like $1/n$.
A: With regard to your question

"what new geometrical phenomena are there in higher dimensions. When I say new I mean phenomena which are counterintuitive or not analogous to their lower dimensional counterparts"

a good example is Borsuk's conjecture, true in dimensions 2 and 3 but false in higher dimensions.  Given a compact pointset $X\subseteq \mathbb R^n$ one defines its diameter to be the maximal distance among a pair of points of $X$. Borsuk conjectured that every such $X$ can be partitioned into at most $(n+1)$ sets of diameter strictly smaller than that of $X$ itself.  For example, the set of vertices of an equilateral triangle in the plane cannot be partitioned into fewer than 3 such sets, and similarly for the set of vertices of a regular $n$-simplex in $\mathbb R^n$.
By playing with figures in the plane it is easy to convince oneself that one can always partition them into subsets of smaller diameter.  In 3-space the problem is trickier but can still be proved.  
It came as a surprise when in 1993 by Jeff Kahn and Gil Kalai proved that the conjecture is false in sufficiently high dimension.  The current best counterexample seems to be in dimension 64.
A: Is the $n$-sphere $S^n\subset \mathbb R^{n+1}$ the underlying real manifold of a complex holomorphic manifold?
Of course this is only possible if $n$ is even, but that is not sufficient.
The real tangent bundle of a complex manifold has the structure of a complex vector bundle and real manifolds whose tangent bundle is endowed with such a complex-linear sructure are called (unsurprisingly!) almost complex manifolds.
The complex-linear tangent bundle structure of an almost-complex manifolds comes from a complex structure of that manifold if and only an easy to check "integrability condition" holds: this is a remarkable theorem of Newlander-Nirenberg.
Now an almost complex structure on $S^n$ only exists for $n=2$ or $6$.
Of course $S^2$ has a complex structure: that of $\mathbb P^1(\mathbb C)$
The other possible sphere $S^6$ is known to have an almost-complex structure, but that structure is not integrable.
So the only remining problem is: does  $S^6$ have another almost complex structure which is integrable, which would imply that $S^6$ has a holomorphic structure.
That problem is generally considered to be  open, despite  purported  "solutions", about which the consensus seems to be that they are insufficiently convincing.
A: The sphere-packing problem, already mentioned in other answers, has some bizarre behavior as the number of dimensions changes. Results in the Leech lattice lead to "monstrous moonshine", and the bizarreness of the sporadic groups. Meanwhile, in string theory, its known that certain things can only happen in 26 or 10 dimensions. Hopf fibrations show you how to deconstruct 3-D space into 2+1 dimensional space, or 11 dimensions into 7+4. But why can't this work in general N, you might wonder? The "homotopy groups of spheres" indicate that even the common-sense notion of a sphere changes from dimension to dimension.
The Laplacian, or rather, the Green's functions which provide the solutions to it, are bizarrely, counter-intuitively different in even and odd-dimensional space, and has the "simplest" form only in 3+1 dimensional space-time (in which we live. Hmmmm. Why would that be?)
A: In high dimensions, almost all of the volume of a ball sits at its surface. More exactly, if $V_d(r)$ is the volume of the $d$-dimensional ball with radius $r$, then for any $\epsilon>0$, no matter how small, you have
$$\lim_{d\to\infty} \frac{V_d(1-\epsilon)}{V_d(1)} = 0$$
Algebraically that's obvious, but geometrically I consider it highly surprising.
Edit:
Another surprising fact: In 4D and above, you can have a flat torus, that is, a torus without any intrinsic curvature (like a cylinder in 3D). Even more: You can draw such a torus (not an image of it, the flat torus itself) on the surface of a hyperball (that is, a hypersphere). Indeed, the three-dimensional hypersphere (surface of the four-dimensional hyperball) can be almost completely partitioned into such tori, with two circles remaining in two completely orthogonal planes (thanks to anon in the comments for reminding me of those two leftover circles). Note that the circles could be considered degenerate tori, as the flat tori continuously transform into them (in much the same way as the circles of latitude on the 2-sphere transform into a point at the poles).
A: A number of problems in discrete geometry (typically, involving arrangements of points or other objects in $\mathbb R^d$) change behavior as the number of dimensions grows past what we have intuition for.
My favorite example is the "sausage catastrophe", because of the name. The problem here is: take $n$ unit balls in $\mathbb R^d$. How can we pack them together most compactly, minimizing the volume of the convex hull of their union? (To visualize this in $\mathbb R^3$, imagine that you're wrapping the $n$ balls in plastic wrap, creating a single object, and you want the object to be as small as possible.)
There are two competing strategies here:


*

*Start with a dense sphere packing in $\mathbb R^d$, and pick out some roughly-circular piece of it.

*Arrange all the spheres in a line, so that the convex hull of their union forms the shape of a sausage.


Which strategy is best? It depends on $d$, in kind of weird ways. For $d=2$, the first strategy (using the hexagonal circle packing, and taking a large hexagonal piece of it) is best for almost any number of circles. For $d=3$, the sausage strategy is the best known configuration for $n \le 56$ (though this is not proven) and the first strategy takes over for larger $n$ than that: the point where this switch happens is called the "sausage catastrophe".
For $d=4$, the same behavior as in $d=3$ occurs, except we're even less certain when. We've managed to show that the sausage catastrophe occurs for some $n < 375,769$. On the other hand, we're not even sure if the sausage is optimal for $n=5$.
Finally, we know that there is some sufficiently large $d$ such that the sausage strategy is always the best strategy in $\mathbb R^d$, no matter how many balls there are. We think that value is $d=5$, but the best we've shown is that the sausage is always optimal for $d\ge 42$. There are many open questions about sausages.

If you're thinking about the more general problem of packing spheres in $\mathbb R^d$ as densely as possible, the exciting stuff also happens in dimensions we can't visualize. A recent result says that the $E_8$ lattice and the Leech lattice are the densest packing in $\mathbb R^8$ and $\mathbb R^{24}$ respectively, and these are much better than the best thing we know how to do in "adjacent" dimensions. In a sense, this is saying that there are $8$-dimensional and $24$-dimensional objects with no analog in $\mathbb R^d$ for arbitrary $d$: a perfect example of something that happens in many dimensions that can't be intuitively described by comparing it to ordinary $3$-dimensional space.

Results like the Hales–Jewett theorem are another source of "new behavior" in sufficiently high-dimensional space. The Hales–Jewett theorem says, roughly speaking, that for any $n$ there is a dimension $d$ such that $n$-in-a-row tic-tac-toe on an $n \times n \times \dots \times n$ board cannot be played to a draw. (For $n=3$, that dimension is $d=3$; for $n=4$, it's somewhere between $d=7$ and $d = 10^{11}$.) However, you could complain that this result is purely combinatorial; you're not doing so much visualizing of $d$-dimensional objects here.
A: If a 2-D plane is perpendicular to the diagonal of a 3-D cube and passes through the midpoint of this diagonal, its intersection with the cube is a regular hexagon.
If a 3-D hyperplane (i.e. given by an equation of the form ax + by + cz + dw = e) is perpendicular to the diagonal of a 4-D hypercube (or tesseract) and passes through the diagonal's midpont, its intersection with the hypercube is a regular octahedron.
I don't think this is very intuitive, but it makes sense after a little thought.
For the 3-D cube has six 2-D faces (squares) and that plane intersects all of them. So, it makes sense that the intersection is a hexagon.
The hypercube has eight 3-D hyperfaces (cubes), so that a hyperplane that intersects them all should do so in an octahedron.
And symmetry suggests the intersections should be regular (although that needs to be proved).
A: I would say 3 dimensions is one of the most interesting dimensions as you have things like knots and 5 platonic solids. The Poincare conjecture for 3-spheres is hard to prove. 
Whereas in higher dimensions tend to get simpler (and more boring/beautiful depending on your point of view.)
Things like the $E_8$ lattice and Leech lattice which solve the sphere packing problems in 8D and 24D are very interesting and are related to many symmetries.
So certain dimensions have these incredible structures but most don't.
Exotic spheres in 4D which I don't even understand!
It's lucky most of the interesting (hard) stuff happens in 3D! Because that's where we live.
A: "What this implies (for me) is the boring consequence that there is no new conceptual richness in higher dimensional geometry beyond the fact than the numbers are larger."
This turns out to be incorrect and should serve as a warning that intuition can lead us astray.
One of the best examples of a completely new property that emerges in higher dimensions involves optimal sphere packing. It transpires that dimensions 8 and 24 are special and different from other dimensions. There exists a lattice in dimensions 8 and 24 which specifies an optimal universal packing:
https://arxiv.org/pdf/1607.02111.pdf
It also turns out that dimension 4 has the unique property that orbits in 4-D space with 3 spatial dimensions are stable, while in higher-dimensional space they are not.
https://physics.stackexchange.com/questions/50142/gravity-in-other-than-3-spatial-dimensions-and-stable-orbits
"In four dimensions, stable orbits are possible. For example, the Earth moves around the sun in the stable orbit, the moon around the Earth in the stable orbit and these are sorts of planar orbits. In higher dimensions, achieving stable orbits is very difficult and generally stable planet orbits don’t exist. And so, it is interesting that somehow Einstein’s equation is telling us that four dimensions are actually the right number of dimensions for life to exists, given that life really depends on the planet moving around the sun and so forth. Four dimensions is really a Goldilocks number. It’s not so too constrained like three dimensions where effectively there is no freedom and no dynamics, but it’s also not too free so that you just have complete chaos, many different solutions, no stable orbits and so on."
http://serious-science.org/higher-dimensional-gravity-6741
Mathematically, certain dimensions associated with symmetry groups have unique properties not found in other numbers of dimensions. For example, the symmetry group E8 (quite famous among mathematicians) describes a space of 248 dimensions. The properties of the E8 Lie Group are unique.  More here:
http://serious-science.org/higher-dimensional-gravity-6741
And here:
https://en.wikipedia.org/wiki/E8_(mathematics)
The group of Clifford algebras has various special properties not found in groups of other size (i.e., dimension):
https://en.wikipedia.org/wiki/Clifford_algebra
Since a given dimension number is generally dual to a Galois group, groups of different size (i.e., dimension) exhibit different mathematical properties. It turns out the group S6 is the only one with a non-trivial outer automorphism, which makes it unique among groups.
https://en.wikipedia.org/wiki/Automorphisms_of_the_symmetric_and_alternating_groups
Topology and groups share a close connection, as do the solution spaces of differential equations and abstract algebras. Different numbers of dimensions have special properties, depending on the abstract algebra or differential equation or topological manifold concerned.  In other words, certain dimensions like 4 or 8 or 24 or 6 exhibit unique properties depending on the situation in the same way that a specific card like an ace can exhibit a special property in the context of the rules of poker.
Many of the answers above gives examples of the unique behavior of various numbers of dimensions, and there exist many more examples throughout mathematics and physics and chemistry and dynamical systems.
