the cup product
Homology groups in different degrees sit side by side with no way to multiply two classes together. Cohomology is richer: it carries a multiplication, the cup product, that combines a class of degree p with a class of degree q to produce one of degree p + q. This turns the graded collection of cohomology groups into a ring, and that ring distinguishes spaces that ordinary homology groups cannot tell apart.
On cochains, the cup product is defined by a front-face/back-face recipe: given an alpha of degree p and a beta of degree q, the cochain alpha cup beta evaluated on a (p+q)-simplex sigma is alpha applied to the front p-face of sigma times beta applied to the back q-face, (alpha cup beta)(sigma) = alpha(sigma restricted to [v_0, ..., v_p]) times beta(sigma restricted to [v_p, ..., v_{p+q}]). One checks this is compatible with the coboundary (a Leibniz rule), so it descends to a well-defined product on cohomology classes H^p(X; R) cross H^q(X; R) -> H^{p+q}(X; R). The result is associative and graded-commutative: alpha cup beta = (-1)^{pq} beta cup alpha, so odd-degree classes anticommute.
Why it matters: the cohomology ring is a strictly finer invariant than the cohomology groups. Two spaces can have isomorphic cohomology groups in every degree yet non-isomorphic rings, and the cup product is what separates them. The classic case is the torus T^2 versus the wedge S^2 wedge S^1 wedge S^1: both have the same Betti numbers, but in the torus the product of the two degree-1 generators is the nonzero generator of H^2, while in the wedge that product vanishes — the ring sees the 'linking' of the two circles that the groups alone miss.
An honest caution: the cup product is natural (a map f induces a ring homomorphism f^* on cohomology), and graded-commutativity carries those signs that are easy to drop. At the cochain level the cup product is associative but not commutative; commutativity only appears after passing to cohomology, and even there it is graded-commutative, not strictly commutative. Homology has no analogous internal product in general — the multiplication genuinely lives on the cohomology side.
On the torus T^2 take the two degree-1 generators alpha, beta of H^1 dual to the two circle factors. Their cup product alpha cup beta is the generator of H^2(T^2) = Z, and beta cup alpha = -alpha cup beta. On the wedge S^2 wedge S^1 wedge S^1 the analogous degree-1 classes cup to zero, so the two spaces have the same groups but different rings.
The cup product separates T^2 from a wedge with identical Betti numbers.
The cup product is only graded-commutative, alpha cup beta = (-1)^{pq} beta cup alpha, so odd classes anticommute — dropping the sign is a common error. Ordinary homology has no comparable internal product; the ring structure is special to cohomology.