the long exact sequence of a pair
Suppose A sits inside X as a subspace. You know the homology of A and you want the homology of X, or vice versa. The long exact sequence of the pair (X, A) is the bookkeeping device that ties together three families of groups — those of A, of X, and of the relative homology H_n(X, A) — into one infinite chain of maps so tight that knowing most of the groups forces the rest.
Relative homology H_n(X, A) is defined from the quotient chain complex C_n(X)/C_n(A): chains in X counted modulo chains already living in A, so it sees the part of X that is genuinely 'new' beyond A. The long exact sequence is then ... -> H_n(A) -> H_n(X) -> H_n(X, A) -> H_{n-1}(A) -> H_{n-1}(X) -> ..., running down forever. The first map is induced by the inclusion A into X, the second by the quotient, and the third — the connecting homomorphism d — takes a relative cycle, applies the boundary, and lands in A (the boundary of a relative cycle is automatically a cycle in A). Exactness means at each group the image of the incoming map equals the kernel of the outgoing one.
Exactness is the whole power: it lets you compute by elimination. If two of three consecutive groups are known, the third is heavily constrained, and where a map is forced to be zero or onto, short exact pieces split off and pin down the unknown group up to extension. This is the single most-used computational engine in homology — you build a space from a subspace, write the sequence, and read off the answer.
An honest subtlety: 'exact' is exactly the condition image = kernel, which is strictly stronger than 'composite is zero' (that only gives image inside kernel). The long exact sequence does not by itself determine groups uniquely when an extension problem appears: from a short exact sequence 0 -> Z -> G -> Z -> 0 you cannot tell whether G is Z + Z without more information. So the sequence constrains, and often determines, but is not a magic black box.
Take X = D^2 the disk and A = S^1 its boundary circle. The disk is contractible so H_n(D^2) = 0 for n > 0 and H_0 = Z. The sequence around degree 2 reads 0 = H_2(D^2) -> H_2(D^2, S^1) -> H_1(S^1) = Z -> H_1(D^2) = 0, forcing H_2(D^2, S^1) = Z. This recovers that the disk relative to its boundary 'is' a 2-sphere homologically.
The pair (D^2, S^1): the LES pins down H_2(D^2, S^1) = Z by elimination.
Exactness means image equals kernel, strictly more than composite-is-zero; and an exact sequence with a nontrivial extension does not determine the middle group up to isomorphism without extra data. Relative homology H_n(X, A) is not the homology of any single space in general — it is the homology of a quotient complex.