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? 

Homeomorphisms are important because they are instances of a more general idea: structure-preserving isomorphisms. You will learn to appreciate this idea as you study more advanced mathematics.
In many domains of mathematical inquiry, the objects of study carry important kinds of "structure," and we don't care to distinguish two objects so long as they have the same structure. We can make the notion of "having the same structure" precise by saying two objects X and Y have the same structure precisely when there is a bijection between them that "preserves the structure" (in a sense that can also be made precise). 
Homeomorphisms are precisely those functions for topology. Their cousins are group isomorphisms in group theory, and ring isomorphisms in ring theory, bijective linear transformations in vector space theory, etc.
