a scheme
A manifold is a space that looks locally like ordinary Euclidean space, even when globally it is something curved and complicated like a sphere or a torus. A scheme is the algebraic-geometry analogue: a space that looks locally like an affine scheme Spec R, glued together along open subsets from many such affine pieces. This single definition unifies number theory and geometry, handles equations over any ring, and carries infinitesimal and arithmetic information that classical varieties simply cannot record.
Precisely, a scheme is a locally ringed space (X, O_X) that admits an open cover by affine schemes: every point has an open neighborhood U such that (U, O_X restricted to U) is isomorphic to (Spec R, O_{Spec R}) for some ring R. You build a scheme exactly the way you build a manifold — take affine charts Spec R_i and glue them along isomorphisms of open subsets, the only constraint being the cocycle compatibility on triple overlaps. A morphism of schemes is a morphism of locally ringed spaces; this is the right notion because the local condition on stalks (maximal ideal to maximal ideal) makes the algebra and geometry match. Standard adjectives sharpen the picture: a scheme is reduced if its rings have no nonzero nilpotents, integral if it is reduced and irreducible (the rings are domains), Noetherian if covered by Spec of Noetherian rings, and one works almost always with morphisms of finite type and with separated schemes (the analogue of Hausdorff).
Schemes are the lingua franca of modern algebraic geometry and arithmetic. Projective space P^n, the moduli spaces, arithmetic surfaces over Spec Z, and every variety are schemes; the framework gives clean fiber products, base change, and a uniform cohomology theory. Several honest cautions distinguish schemes from varieties. A scheme can be NON-reduced — the structure sheaf may contain nilpotents, encoding infinitesimal thickening like a 'double point' Spec k[x]/(x^2), which is invisible as a set but real as a scheme. A scheme need not be over an algebraically closed field, or over a field at all (Spec Z is a scheme), so the variety-level Nullstellensatz does not govern it. And the Zariski topology of a scheme is coarse and usually non-Hausdorff, with generic points; intuition trained on Hausdorff manifolds must be adjusted. The genuine content of a scheme lives in its structure sheaf, not in its underlying topological space.
The projective line P^1 over a field k is the simplest non-affine scheme: glue two copies of the affine line A^1 = Spec k[x] and Spec k[y] along the open subsets where x and y are nonzero, via the relation y = 1/x. The result is covered by two affine charts but is not itself the Spec of any single ring — its only global regular functions are the constants k, just as on the compact Riemann sphere.
P^1: two affine lines glued by y = 1/x — a scheme, but not an affine one.
Unlike a variety, a scheme may be non-reduced (nilpotents in O_X) and need not be over an algebraically closed field — the Nullstellensatz is a variety-level statement. The real data is the structure sheaf, not the (coarse, non-Hausdorff) point set.