scheme
A scheme is to algebraic geometry what a manifold is to differential geometry: a global object built by gluing simple local charts. The charts here are prime spectra of commutative rings, and gluing them lets one do geometry over any ring at once — over the integers, over finite fields, even over rings with nilpotents that encode infinitesimal thickening.
Precisely, a scheme is a locally ringed space (X, O_X) that is locally isomorphic, as a locally ringed space, to the spectrum Spec R of some commutative ring together with its structure sheaf. The affine schemes Spec R are the local models; a general scheme is covered by open sets each isomorphic to an affine scheme, and the structure sheaf records the regular functions, whose stalks are local rings.
Schemes vastly enlarge the classical theory. They admit nilpotents, so a 'doubled point' Spec k[x]/(x^2) is a genuine nonreduced object remembering a tangent direction; they have generic points making irreducibility transparent; and they support fibre products, base change, and a clean functor of points. Varieties reappear as the reduced, separated, finite-type schemes over a field, so nothing classical is lost while infinitely much is gained — the framework in which modern arithmetic and algebraic geometry are written.
Spec k[x]/(x^2) is a one-point space, but its ring has the nilpotent x; it is the 'doubled origin' or first-order infinitesimal neighborhood, and maps out of it into a variety detect tangent vectors.
A nonreduced one-point scheme — invisible classically, but it carries a tangent direction.
Historically the term 'scheme' was Grothendieck's; what older authors called a 'scheme' (Chevalley) or 'prevariety' differs. Today a scheme need not be separated or reduced, and a 'variety' usually means an integral separated scheme of finite type over a field.