an affine scheme
An affine scheme is the geometric incarnation of a single commutative ring. It is the most basic kind of scheme — the atom from which all others are built — and it is, by definition, just the prime spectrum Spec R of a ring R, equipped with its structure sheaf O_{Spec R}. The slogan is that the category of affine schemes is an exact mirror of the category of commutative rings, run backwards: rings and their geometry are literally the same information viewed from two sides.
Precisely, an affine scheme is a locally ringed space (X, O_X) that is isomorphic to (Spec R, O_{Spec R}) for some commutative ring R; the ring R is recovered as the global sections O_X(X). The defining theorem of the subject is an anti-equivalence of categories: the functor R -> Spec R is a contravariant equivalence between commutative rings and affine schemes, with inverse 'take global sections'. Concretely this means a morphism of affine schemes Spec S -> Spec R is THE SAME DATA as a ring homomorphism R -> S, going the other way — there is no loss and no extra choice. This is why algebra and geometry can be translated freely: an ideal of R becomes a closed subscheme, a quotient R -> R/I becomes a closed immersion, a localization R -> R_f becomes an open immersion onto D(f), and the tensor product of rings becomes the fiber product of schemes.
Affine schemes are the local models for all of geometry the way Euclidean balls are the local models for manifolds: a general scheme is a space that looks locally like an affine scheme, glued from Spec's. Two honest cautions. First, 'affine' here means 'glued from one ring', and most interesting schemes — projective space CP^n above all — are NOT affine; affineness is a strong, special restriction (Serre's criterion: X is affine iff all higher cohomology of every quasi-coherent sheaf vanishes). Second, an affine scheme can be far weirder than a classical affine variety: Spec R is defined for any ring, so it may be non-reduced (Spec k[x]/(x^2), a doubled point), non-Noetherian, or over a field that is not algebraically closed, where the Nullstellensatz dictionary of Algebraic Geometry I no longer applies.
Affine n-space over a field k is A^n_k = Spec k[x_1, ..., x_n]; the ring map k[x] -> k[x]/(x^2 - x) (sending the polynomial relation to zero) corresponds geometrically to the closed immersion of the two points {0, 1} into the affine line. The anti-equivalence is at work: quotient on the ring side, closed subscheme on the geometry side.
Rings and affine schemes are anti-equivalent: a quotient ring is a closed subscheme.
Most schemes are NOT affine — projective space is the basic non-example. Affineness is the strong condition that all higher cohomology of quasi-coherent sheaves vanishes (Serre's criterion), not a default.