Why are homeomorphisms important? I attended a guest lecture (I'm in high school) hosted by an algebraic topologist. Of course, the talk was non-rigorous, and gave a brief introduction to the subject. I learned that the goal of algebraic topology is to classify surfaces in a way that it is easy to tell whether or not surfaces are homeomorphic to each other. I was just wondering now, why are homeomorphisms important? Why is it so important to find out whether two surfaces are homeomorphic to each other or not? 

Just to add to the already-good answers provided so far: Human beings in general, and scientists in particular, are classifiers.  We group "like things" in buckets and try to make statements that must be true for everything in the bucket (e.g., how biologists classify kingdoms, phyla, species, etc.)  In the case of topology, we can place spaces in buckets according to their homeomorphism type.  We could, instead, use other classification schemes (e.g., homotopy type, which is a looser form of equivalence.)
Category theory provides a nice setting for making sense of this general, scientific process: When studying a certain class of objects, the fundamental (e.g., "important"/"interesting"/primary) goal is to classify the objects up to categorical isomorphism.  And for the category of topological spaces and continuous maps, isomorphisms are homeomorphisms.
