the spectrum of a ring
Classical algebraic geometry studies the solution set of polynomial equations, and over an algebraically closed field these solutions correspond to maximal ideals of the coordinate ring. Grothendieck's revolutionary move was to take ANY commutative ring R and manufacture a geometric space out of it directly, using not just its maximal ideals but all its prime ideals as the points. The result, Spec R, is the universal 'space of the ring': every commutative ring, no matter how abstract — the integers Z, a polynomial ring, a ring with nilpotents — becomes a geometric object you can do topology and sheaf theory on.
Precisely, the prime spectrum Spec R is the set of all prime ideals p of R, made into a topological space by the Zariski topology: the closed sets are V(I) = {primes containing the ideal I}, one for each ideal I of R. Equivalently, the open sets are generated by the distinguished opens D(f) = {primes not containing f}, for elements f of R; D(f) is where 'the function f is nonzero'. A point of Spec R is a prime ideal, and you think of it as a (possibly fat) subvariety: the maximal ideals are the classical closed points, while a non-maximal prime like (0) in an integral domain is a generic point whose closure is the entire space. Each element f of R is reinterpreted as a function on Spec R: its 'value' at a prime p is its image in the residue field k(p) = Frac(R/p), and f vanishes at p exactly when f lies in p. The construction is contravariantly functorial: a ring map R -> S induces a continuous map Spec S -> Spec R by pulling primes back.
Spec R is the building block of all of scheme theory — every scheme is glued from pieces of the form Spec R — and it sees structure that classical varieties cannot. Because primes, not just maximal ideals, are points, Spec R has generic points and a genuine notion of irreducible components (V(p) for minimal primes p). Because R can have nilpotents, Spec R can carry infinitesimal thickening invisible at the level of the underlying set. Two honest cautions: first, the Zariski topology is extremely coarse and almost never Hausdorff — a generic point is dense, so 'points are close together' is the norm, not the exception. Second, an element of R is NOT an ordinary function: its values at different points can live in different fields (Z has F_p-valued behavior at the prime (p) and Q-valued behavior at (0)), so 'f as a function on Spec R' is a function whose codomain varies from point to point.
Spec Z, the spectrum of the integers, has one point (p) for each prime number 2, 3, 5, ... — these are the closed points, with residue field F_p — plus the generic point (0), with residue field Q, whose closure is all of Spec Z. An integer n, viewed as a function, has 'value' n mod p at the point (p): so 6 vanishes at (2) and (3). This is the precise sense in which number theory becomes geometry.
Spec Z: a closed point per prime (residue field F_p) plus a dense generic point (0) with residue field Q.
Points of Spec R are PRIME ideals, not just maximal ones, so there are generic (non-closed) points; the topology is rarely Hausdorff. And an element of R is a 'function' whose values live in different residue fields at different points.