integral closure
When a ring has 'missing' elements that morally ought to belong to it — fractions that satisfy honest monic equations but were left out — the integral closure adds them all back in. Geometrically this is normalization: it repairs the mild singularities of a curve, separating self-crossings and unpinching cusps, replacing a flawed model by its smoothest birational companion. It is the cleanup operation that makes a ring 'integrally closed'.
Let R be a subring of a commutative ring S. The integral closure of R in S is the set of all elements of S that are integral over R, i.e. that satisfy a monic polynomial with coefficients in R. A nontrivial theorem ensures this set is itself a subring containing R. When R is an integral domain and S = Frac(R), one speaks of the integral closure of R, and R is called integrally closed (or normal) if it equals its own integral closure in its fraction field.
Every unique factorization domain is integrally closed, so Z and k[x_1, ..., x_n] are normal; this is why 'normal' is a weakening of 'UFD'. The classic non-normal example is Z[sqrt(-3)], whose integral closure in Q(sqrt(-3)) is the larger ring Z[(1 + sqrt(-3))/2] — the element (1 + sqrt(-3))/2 satisfies the monic x^2 - x + 1 = 0 and so was integral but missing. For finitely generated domains over a field the integral closure is again finitely generated (the Noether finiteness theorem), making normalization computable.
The cuspidal cubic ring R = k[t^2, t^3] ⊆ k[t] has integral closure k[t]: the element t = t^3/t^2 lies in Frac(R) and satisfies the monic x^2 - t^2 = 0 over R, so normalization separates the cusp into a smooth line.
Normalization resolves the cusp of y^2 = x^3.
The ring of integers of a number field is, by definition, the integral closure of Z in that field — integral closure is precisely the mechanism that produces Dedekind domains in number theory.