What is the geometry in algebraic geometry? Coming from a physics background, my understanding of geometry (in a very generic sense) is that it involves taking a space and adding some extra structure to it. The extra structure takes some local data about the space as its input and outputs answers to local or global questions about the space + structure. We can use it to probe either the structure itself or the underlying space it lives on. For example, we can take a smooth manifold and add a Riemannian metric and a connection, and then we can ask about distances between points, curvature, geodesics, etc. In symplectic geometry, we take an even-dimensional manifold and add a symplectic form, and then we can ask about... well, honestly, I don't know. But I'm sure there is interesting stuff you can ask.
Knowing very little about algebraic geometry, I am wondering what the "geometry" part is. I am assuming that the spaces in this case are algebraic varieties, but what is the extra structure that gets added? What sorts of questions can we answer with this extra structure that we couldn't answer without it?
I have to guess that this is a little more complicated than just taking a manifold and adding a metric, otherwise I would expect to be able to find this explained in a relatively straightforward way somewhere. If it turns out the answer is "it's hard to explain, and you just need to read an algebraic geometry text," then that's fine. In that case, it would be interesting to try to get a sense of why it's more complicated. (I have a guess, which is that varieties tend to be a lot less tame than manifolds, so you have to jump through more technical hoops to tack on extra stuff to them, but that's pure speculation.)
 A: This is a big complicated question and many different kinds of answers could be given at many different levels of sophistication. The very short answer is that the geometry in algebraic geometry comes from considering only polynomial functions as the meaningful functions. Here is essentially the simplest nontrivial example I know of:
Consider the intersection of the unit circle $\{ x^2 + y^2 = 1 \}$ with a vertical line $\{ x = c \}$, for different values of the parameter $c$. If $-1 < c < 1$ we get two intersection points. If $c > 1$ or $c < -1$ we get no (real) intersection points. But something special happens at $c = \pm 1$: in this case the vertical lines $x = \pm 1$ are tangent to the circle. This tangency is invisible if we just consider the "set-theoretic intersection" of the circle and the line, which consists of a single point; for various reasons (e.g. to make Bezout's theorem true) we'd like a way to formalize the intuition that this intersection has "multiplicity two" in some sense, and so is geometrically more interesting than just a single point.
This can be done by taking what is called the scheme-theoretic intersection. This is a complicated name for a simple idea: instead of asking directly what the intersection is, we ask what the ring of polynomial functions on the intersection is. The ring of polynomial functions on the unit circle is the quotient ring $\mathbb{R}[x, y]/(x^2 + y^2 - 1)$, while the ring of polynomial functions on the vertical line is the quotient ring $\mathbb{R}[x, y]/(x - c) \cong \mathbb{R}[y]$. It turns out that the ring of polynomial functions on the intersection is the quotient by both of the defining polynomials, which gives, say at $x = 1$ to be concrete,
$$\mathbb{R}[x, y]/(x^2 - y^2 - 1, x - 1) \cong \mathbb{R}[y]/y^2.$$
This is a funny ring: it has a nontrivial nilpotent! That nilpotent $y$ is exactly telling us the sense in which the intersection has "multiplicity two"; it's saying that a function on the scheme-theoretic intersection records not only its value at the intersection point but its derivative with respect to tangent vectors at the intersection point, reflecting the geometric fact that the unit circle and the line are tangent and so share a tangent space. In other words it is saying, roughly speaking, that the intersection is "two points infinitesimally close together, connected by an infinitesimally short vector."
Adding nilpotents to geometry takes some getting used to but it turns out to be very useful; among other things it is possible to define tangent spaces in algebraic geometry this way (Zariski tangent spaces), hence to define Lie algebras of algebraic groups in a purely algebraic way.
So, this is one story you can tell about what kind of geometry algebraic geometry captures, and there are many others, for example the rich story of arithmetic geometry and its applications to number theory. It's difficult to say anything remotely complete here because algebraic geometry is absurdly general and the sorts of geometry it is capable of capturing veer off in wildly different directions depending on what you're interested in.
A: A classical manifold is a space that locally looks like $\mathbb{R}^n$; or, via results like the Whitney embedding theorem, a suitably nice subspace of some $\mathbb{R}^N$. If "looks like" involves some notion of smoothness, for example, then we can expand into differential geometry and talk about constructions like tangent spaces and differential forms. If we stick to just continuity, then we can still work with some constructions like homology and cohomology (just not, say, de Rham cohomology), and we can deal with more pathological spaces and functions between them.
A natural question to ask, then, is what's so special about $\mathbb{R}^n$? We can consider spaces that locally look like an arbitrary Banach space, for example. (I don't think this is a particularly popular approach, at least, at the undergrad/early grad school level, but Abraham, Marsden, and Ratiu works in this category.) The starting point of algebraic geometry is wanting to deal with spaces over an arbitrary commutative ring. It's not clear how continuity or smoothness should map over to this case, but at the very least polynomials make sense over an arbitrary ring, and we can look at the space like $V(f) = \{x\in k^n:\, f(x) = 0\}$ for a polynomial $f\in k[X_1, \dots, X_n]$. But that's not exactly what we want either; for the important case of $k$ finite, for example, $V(f)$ is just a finite collection of points.
The analogy that turns out to work is going in the opposite direction, and trying to generalize the idea of functions on a manifold. To that end, algebraic geometry works with locally ringed spaces, which are pairs $(X, \mathcal{O}_X)$ with $X$ a topological space and $\mathcal{O}_X$ a sheaf of rings on $X$ satisfying properties roughly analogous to what you'd expect for, say, smooth functions on a manifold. In rough terms, what you wind up with is a space that locally looks like the spectrum of a commutative ring--- but unlike the case of real manifolds, that ring can vary along the space. That's admittedly a vague analogy, and it takes a lot of technical results to even talk about the resulting object. But if you're familiar with vector bundles, for example, then consider Swan's Theorem: For a smooth, connected, closed manifold $X$, the sections functor $\Gamma(\cdot)$ gives an equivalence between vector bundles over $X$ and f.g., projective modules over the ring $C^\infty(X)$.
So, what makes this algebraic thing we've constructed look geometric? Smoothness doesn't make sense outside of $\mathbb{R}^n$, but if we're working with polynomials, they have a formal derivative that allows us to do roughly the same thing. More generally, a local ring $(R, \mathfrak{m})$ has a cotangent space $\mathfrak{m}/\mathfrak{m}^2$ that's roughly analogous to the cotangent space of a manifold; and with a bit of work, we can get something that at least has some of the formal properties one wants for a tangent or cotangent space. Even though the topology we're working with turns out to be much more complicated than the case of manifolds (the Zariski topology, for example, is generally non-Hausdorff), we still have a notion of cohomology (the simplest being Cech cohomology with a sheaf). There's a massive jump in abstraction and technical requirements compared with the more geometric case, but algebraic geometry turns out to be the right extension of more familiar geometry when dealing with things such as, say, number fields.
