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A homeomorphism is a continuous bijection with continuous inverse. Two spaces are homeomorphic if such a map exists; they are topologically identical. Homeomorphisms preserve all topological properties: compactness, connectedness, dimension, fundamental groups. Classification of topological spaces is the problem of describing spaces up to homeomorphism.
From the topological definition of continuity, you know that a continuous function f: X → Y "respects" the topology: preimages of open sets in Y are open in X. But continuity alone, even with bijectivity, does not make f a topological equivalence. A homeomorphism adds the requirement that the inverse f⁻¹ is also continuous — so the topology flows both ways, and open sets in X correspond exactly to open sets in Y. A homeomorphism is the topology's notion of "being the same."
A classic example reveals why the inverse's continuity matters. Consider the map f: [0,1) → S¹ defined by f(t) = (cos 2πt, sin 2πt) — wrapping the half-open interval onto the unit circle. This is a continuous bijection, but f⁻¹ is not continuous (small open sets near the "seam" of the circle pull back to disconnected sets near 0 and 1). So f is not a homeomorphism: [0,1) and S¹ are topologically distinct. The circle is compact; the half-open interval is not — and compactness is a topological property that homeomorphisms must preserve.
Topological properties are precisely those that homeomorphisms preserve: compactness, connectedness, path-connectedness, the number of connected components, the fundamental group, and dimension. If X and Y are homeomorphic, they must agree on all of these. This turns homeomorphism classification into a game: to show two spaces are homeomorphic, exhibit a homeomorphism; to show they are not, find a topological property they disagree on. The circle S¹ and the interval [0,1] both have one connected component, but removing a point from S¹ leaves the space connected while removing an interior point from [0,1] disconnects it — so they are not homeomorphic.
The popular analogy — a topologist cannot tell a coffee mug from a donut — captures this precisely. Both have exactly one "hole" (they are homeomorphic to each other and to S¹ × D²), and any property topology can detect, they share. The program of classifying spaces up to homeomorphism is one of topology's central ambitions: the classification of compact surfaces (sphere, torus, Klein bottle, …) and the ongoing program of understanding three-manifolds are examples at different levels of complexity. Every theorem you will prove about continuous functions on topological spaces — quotient maps, the Tietze extension theorem — depends on understanding when two spaces are genuinely different versus merely described differently.