The excision theorem states that if Z is a subspace of A whose closure is contained in the interior of A, then the inclusion (X \ Z, A \ Z) -> (X, A) induces isomorphisms H_n(X \ Z, A \ Z) = H_n(X, A) for all n. In other words, we can "cut out" (excise) the subspace Z from both X and A without changing the relative homology. This theorem is what gives homology its local-to-global computational power: relative homology depends only on the behavior near the boundary of A in X, not on what happens deep inside A or far from A.
The excision theorem is one of the Eilenberg-Steenrod axioms for homology theories, and it is the property that gives homology its remarkable computational power. The precise statement: if (X, A) is a pair and Z is a subset of A with cl(Z) contained in int(A), then the inclusion map (X \ Z, A \ Z) hookrightarrow (X, A) induces isomorphisms H_n(X \ Z, A \ Z) -> H_n(X, A) for all n. Equivalently (and often more useful in practice): if X = A union B with int(A) and int(B) covering X, then the inclusion (B, A intersect B) hookrightarrow (X, A) induces isomorphisms H_n(B, A intersect B) -> H_n(X, A).
The intuition behind excision is that relative homology H_n(X, A) measures the topology of X in the neighborhood of the "boundary" between A and its complement X \ A. What happens deep inside A is invisible (it is already quotiented out), and what happens far from A in X contributes nothing to the relative chains (which must have boundaries in A). Therefore, cutting out a piece Z that is buried deep inside A has no effect on the relative homology. The formal proof uses the technique of barycentric subdivision: singular chains can be subdivided into smaller and smaller pieces until every singular simplex maps either entirely into A or entirely into B = X \ Z, allowing chains to be decomposed into local contributions. This subdivision process does not change homology (subdivided chains are homologous to the originals), and once chains are local, the excision isomorphism follows.
The most important consequence of excision is the identification H_n(X, A) = reduced H_n(X/A) for good pairs (pairs where A is a neighborhood deformation retract in X). The proof: let X/A be the quotient space obtained by collapsing A to a point p. Then H_n(X/A, p) = H_n(X/A) for n > 0 (since a point has trivial higher homology). Excision (in the equivalent formulation) identifies H_n(X, A) with H_n(X/A, A/A) = H_n(X/A, p). This identification makes relative homology geometric: H_n(X, A) measures the holes in X that survive when we crush A to a point.
Excision is the engine behind the Mayer-Vietoris sequence. Given X = A union B with open cover, the long exact sequence of the pair (X, A) involves H_n(X, A). Excision identifies H_n(X, A) with H_n(B, A intersect B) (excise Z = X \ B). Substituting this into the long exact sequence and rearranging gives the Mayer-Vietoris sequence: ... -> H_n(A intersect B) -> H_n(A) direct sum H_n(B) -> H_n(X) -> H_{n-1}(A intersect B) -> ... This sequence allows computation of H_n(X) from the homology of the pieces A, B, and their intersection. Similarly, excision underlies cellular homology: the relative group H_n(X^n, X^{n-1}) for a CW complex is computed by excising the (n-2)-skeleton, reducing to a wedge of spheres and giving H_n(X^n, X^{n-1}) = Z^{number of n-cells}. Without excision, neither of these fundamental computational tools would exist.