higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
In algebraic geometry, algebraic variety (not to be confused with variety of algebras) is a scheme which is integral, separated? and of finite type over an algebraically closed field $k$.
Classically, the term algebraic variety referred to a scheme as above which is further quasi-projective, i.e. admits a locally closed embedding into projective space. Thus, these were objects which locally are cut out inside projective space as the geometric locus of zeros of a set of polynomial equations in finitely many variables. (The first example of an algebraic variety which is not quasi-projective was given by Nagata?.)
Historically, there were several formalisms of various schools including the Italian school of algebraic geometry in the early 20th century (Veronese, Castelnuovo, Severi, …), the American school between the two wars (Oscar Zariski?), Andre Weil), the abstract varieties of Jean-Pierre Serre and finally the language of schemes introduced by the Grothendieck school. One should note that in the case of (esp. projective) varieties over complex numbers there is an additional possibility to work using complex-analytic tools and complex topology.
Given an algebraically closed field $k$, an algebraic $k$-variety usually means either a quasiprojective variety or an abstract variety (in the sense of Serre). ‘Quasiprojective’ unifies affine, quasiaffine, projective and embedded quasiprojective $k$-varieties. Many modern sources by a variety mean a reduced separated scheme of finite type over a field, often requiring also irreducibility (that is integral = reduced and irreducible).
An embedded affine $k$-variety (or an affine algebraic set) is a set of zeros of a locus of common zeros of a set of polynomial equations in the affine space $\mathbf{A}^n_k$. By the Hilbert Nullstellensatz there is a more invariant definition. Affine $k$-varieties are maximal spectra (= sets of maximal ideals) of finitely generated noetherian (commutative unital) $k$-algebras without nilpotents with the Zariski topology; the algebra can be recovered as the coordinate ring of the variety; this correspondence is an equivalence of categories, if the morphisms are properly defined.
Affine varietes can be embedded as closed subvarieties into an affine space (in the sense of algebraic geometry). As topological spaces affine varieties are noetherian.
Projective $k$-varieties are obtained in a similar way from graded $k$-algebras, or, in embedded incarnation, as loci of zeros of a set of homogeneous polynomials in projective space $\mathbf{P}^n_k$.
Embedded quasiaffine $k$-varieties are Zariski-open subspaces of affine $k$-varieties.
Embedded quasiprojective $k$-varieties are Zariski-open subspaces of projective $k$-varieties. We can remove the embedding by equipping them with the sheaf of regular functions and therefore considering them as locally ringed spaces. In the category of locally ringed spaces, projective, affine, and quasiaffine varieties are (isomorphic to) special cases of quasiprojective. Alternatively, we can put all 4 classes without sheaves into a category, by defining regular maps directly, and we get an isomorphic category of varieties.
In fact, by noticing that the affine $k$-space is Zariski open in a projective space of the same dimension, we see that the quasiprojective case includes all others.
Morphisms between varieties are so-called regular maps?.
Sometimes a smooth algebraic variety may also be called algebraic manifold.
An abstract $k$-prevariety in the sense of Serre is a locally ringed space which is locally isomorphic to affine $k$-variety. The category of $k$-prevarieties has a product which is obtained by locally gluing products in the category of affine $k$-varieties. This enables defining a diagonal $X\to X\to X$; a prevariety is separated, or an abstract $k$-variety if the diagonal is closed in Zariski topology (which is, of course, not a product of Zariski topologies of factors).
There is an equivalence of categories between the category of integral schemes of finite type over $Spec\,k$, where $k$ is an algebraically closed field, and the category of (irreducible) algebraic $k$-varieties.
Of course, given a variety the corresponding scheme and variety have different sets of points; the points in common are the closed points of the scheme. The remaining points are the generic points of subvarieties. Generic points were often used, without proper foundations, in other language, already in the works of the Italian school.
Some modern algebraic geometers mean, by varieties, objects of certain slightly bigger categories of relative $S$-schemes of finite type (where $S$ is not necessarily $Spec\,k$ for $k$ a field); typically they are required to be separated reduced $S$-schemes of finite type.
An amusing discussion on the differences between schemes and varieties can be found at Secret blogging seminar: algebraic geometry without prime ideals.