Topology is the study of shape
It is the most exciting subject I've ever heard of, but that may be due to my never having attended an actual lecture on it.
Why is topology exciting? It all depends if you're attracted to pure deductive reasoning. Like Euclidean geometry, topology is defined in terms of a few elementary propositions about elementary concepts. Like Euclid's axioms of geometry, the axioms of topology capture a real intuition of us humans about shape and appearance, a certain sense we have of when two shapes are "fundamentally the same" and the theorems we can derive in topology therefore tell us something about the real world, or at least, about deep intuitions we humans have about this world.
In Euclidean geometry, objects have distance; in topology, they don't. In Euclidean space, two objects are "fundamentally the same" when they have the same shape, i.e. when one can be transformed into the other by movement (translation, rotation) but without changing any distances within the objects; in topology, it is also permissible to stretch, shrink and bend them into each other, without disrupting the surface - objects are "made out of dough".
This can also be formulated by saying: topology is the study of connectedness, of the differences in ways that things can be put together. A topology is a particular way in which something is put together. This is also a term used quite often outside the mathematical field of topology - see Rancid_Pickle's writeup below.
Any abstract mathematical theory can bring the same sense of excitement: as soon as I feel the intuitions that it captures, the formulas come to life, and working with them becomes an adventure of discovery. Geometry and topology just happen to have an edge because they are simple and the intuitions they deal with apply to physical objects. This is why Newtonian mechanics has a similar appeal, or, in a very different area, the theory of Turing machines. It is much harder for quantum mechanics or process algebra to capture the imagination.