What exactly are the properties of Topology? I've looked through quite a few resources on the internet and I can't really find a good answer to this question. I am just starting to get into topology and manifolds and I came across the statement "A coffee mug and a torus are topologically the same....any closed loop and a circle are topologically the same."
I mean obviously, they are geometrically pretty different. So what are the properties that topology is specifically concerned with? I know manifolds are a complete different story and give us much greater insight into the geometry of our given shape, but I'm specifically wondering what topology "measures." A mathematical answer and a layman answer would be greatly appreciated, thanks!
(if this is too general of a question, maybe give the 3 most important properties)
 A: Topology is concerned with continuous functions.  There is a continuous function by which points on the surface of the coffee cup correspond to points on the surface of the donut.  It is "continuous in both directions", i.e. if $f$ goes from the cup to the donut, then $f$ and $f^{-1}$ are both continuous.  Intuitively, you can change one surface into the other by stretching without tearing.
Consider the function from $[0,2\pi)$ to the circle, parametrized by $\theta\mapsto(\cos\theta,\sin\theta)$.  That's a continuous function, but its inverse is not continuous, since as you go around the circle, $\theta$ makes a discontinuous jump from one end of the interval to the other as the point on the circle goes past $(1,0)$.  But now let's alter which subsets of $[0,2\pi)$ we consider to be open sets: a set containing $0$ will be open only if it includes as a subset $[0,\varepsilon)\cup(2\pi-\varepsilon,2\pi)$ for some $\varepsilon>0$.  Every open set containing $0$ covers parts of both ends.  Then we have in effect glued the ends together, and the jump from one end of the interval to the other is no longer a discontinuous jump.  Suddenly the function in one direction and the function in the other direction are both continuous.  One moral is this: the topology on the space is just a matter of which sets are considered to be open.
A: To add to Henry's answer, topology is concerned with understanding topological spaces, which are a very general class of objects that massively generalize things like circles, donuts, coffee cups, higher dimensional objects, etc.
Topological spaces come with just enough information to talk about continuity, and we can talk about spaces being compact, which means closed and bounded, like a solid ball, or connected, meaning it can't be split into disjoint pieces. So for example, the letter X is connected, but a lowercase i is not. 
One way we try to understanding these spaces is by associating simple algebraic objects to them, such as groups. This is called algebraic topology. This is where Henry Swanson's fundamental group lives, as well as other objects called cohomology and homology groups, and other objects called homotopy groups which generalize the fundamental group. These objects are pretty abstract, and some are more easily computable than others. 
So, how do these algebraic objects help us? Well if two spaces don't have the same object associated with them, then they are not the same in some technical sense. It is important to know that there are many senses in which objects can be the same or not, i.e. homeomorphic, homotopy equivalent, etc.
What are these objects measuring? This is pretty interesting. The fundamental group measures the "holes" an object has. Sometimes nonzero cohomology groups represent an obstruction for certain equations to gave solutions. The list goes on...
A: I'm awful with topology, so this is mostly a layman's answer.
In low dimensions ($3$ and lower), the fundamental group is good for understanding the behavior of a space. It measures how loops behave on a surface (or $n$-volume, but 2D is easy to picture). Imagine you have an ant on the surface of a sphere. It anchors the rope to a point, then wanders around the sphere until it comes back to its starting point, forming a loop. If it pulls on the rope, the loop will contract down to a point. But this is not true if the ant lives on a torus, the loop can get "caught" on the hole in the middle, or around the doughy part. Under concatenation, these loops form a group (for the torus it's $\mathbb{Z}^2$). But in dimensions $4$ and higher, you can construct manifolds that have any group as their fundamental group, so it becomes much less useful.
On the plus side, in dimensions $5$ or higher, many things simplify, because you can perform surgery on them. I'm not quite sure what the deep questions in this area are, but it seems neat to snip things apart and glue them back together.
Then there's point-set topology, which just baffles me. It's disturbingly general; you start with just a set, and a definition of which subsets are "open", and you then get continuity and limits out of that somehow. I'm pretty sure that whether spaces are compact is a very important thing, but I missed that mini-lecture.
A: For a much shorter answer - topology is concerned with global structure of shapes, whereas geometry is concerned with the small-scale structure. This difference is exemplified by your doughnut versus coffee mug example; "they both have one hole" is the large-scale structure, but at small scales the doughnut and coffee mug look quite different.
