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.

The notion of homeomorphism is of fundamental importance in topology because it is the correct way to think of equality of topological spaces. That is, if two spaces are homeomorphic, then they are indistinguishable in the sense that they have exactly the same topological properties.

If they are homeomorphic, they have the same topological properties. Topologically, they are the same, thus the joke that a topologist cannot tell apart the doughnut from the coffee mug. They are the same. Intuitively, you can transform a surface into another without making any tearings in the surface. If your doughnut is a muffin, without a hole, you cannot transform it into the coffee mug (or the doughnut) without making a hole, thus breaking the structure. The muffin is not homeomorphic to the coffee mug, that's why you should never have muffins with your coffee.

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.

The specific form of the 20th century definition of homeomorphism, with inverse pairs of continuous functions, is not necessarily important. It was not originally used for the classification of shapes of 2-dimensional surfaces, a problem solved in the middle of the 19th century without a precise definition of "surface" or (topological) "shape". Some other shape classification problems, such as determining which knots can be untied, have never used homeomorphism as the definition of equivalence.

What is more important than the particular definitions, is to understand, classify and inter-relate the shapes of geometric objects, and to have a theory of that can give clear formulations of all those problems and provide tools for solving them.

If "shape" is formalized in terms of the formal data used in topology (open sets, continuous maps, convergence, ...) then homeomorphism is by definition the notion of equivalence associated to that; a one-to-one correspondence that matches the data. When the lecturer equated the goal of studying shapes with the goal of studying spaces up to homeomorphism, that was the same as saying that people have settled on the theory of topological spaces as an adequate formalization of "shape".