Algebraic geometry is, in origin, a geometric study of solutions of systems of polynomial equations and generalizations.
The set of zeros of a set of polynomial equations in finitely many variables over a field is called an affine variety and it is equipped with a particular topology called Zariski topology, whose closed sets are subvarieties. The system of polynomial equations defines an ideal in the ring of polynomials over the ground field; one of the first insights of algebraic geometry is that the ideal is a more invariant notion than the original set of equations. The quotient of the ring of polynomials by the defining ideal is the ring of coordinate functions of the affine variety; a basic theorem asserts that this ring is Noetherian algebra over the ground field. This ring determines the variety up to a natural notion of isomorphism of varieties. Projective varieties are zeroes of systems of homogeneous polynomial equations in projective $n$-dimensional space; if the coordinates are rescaled by a common nonzero multiple, then by definition they still define the same point in projective space. Zariski open subsets of affine (or projective) varieties are called quasiprojective varieties.
It is often useful to take a pullback along a morphism of fields, usually called base change. Since Grothendieck, one generalizes the coordinate rings of affine varieties to arbitrary commutative unital rings, not necessarily Noetherian nor finitely generated; and interprets the opposite category of the category of commutative rings as a category of affine schemes $\mathrm{Aff}$; affine schemes are traditionally constructed by the affine spectrum functor $\mathrm{Spec}:\mathrm{CommRing}^{\mathrm{op}}\to\mathrm{lrSp}$ into the category of locally ringed topological spaces. Points of the affine spectrum are the prime ideals of the ring. Assuming the axiom of choice, every affine scheme corresponding to a ring with $0 \neq 1$ has therefore at least one point. The slice category $\mathrm{Aff}/(\mathrm{Spec} F)$ over a spectrum of a fixed field $F$ contains the category of varieties over $F$ as a full subcategory. However the points of an affine variety and of the corresponding affine spectrum do not coincide: only maximal ideals are points of usual varieties. Affine schemes have a natural topology, also called a Zariski topology; the ringed spaces locally isomorphic to affine schemes are called schemes. Schemes include projective schemes and more generally quasiprojective schemes; if they are relative over a fixed ground field, then they contain the subcategories of projective (resp. quasiprojective) varieties over the same field.
Although Grothendieck in the late 1950s envisioned many generalizations of scheme theory, his coauthor Dieudonné wrote later in EGA that algebraic geometry is the study of algebraic and formal schemes, which is clearly a too dogmatic definition. Grothendieck’s school studied in addition locally affine spaces in various Grothendieck topologies on $\mathrm{Aff}$ (including algebraic spaces), algebraic stacks (Deligne-Mumford stacks and Artin stacks), ind-schemes and so on; in SGA the study of ringed spaces is replaced by more general ringed sites and ringed topoi. Modern generalizations include derived schemes, almost schemes (with the theory of almost rings of Gabber developing after some ideas of Faltings), generalized schemes of Nikolai Durov, so-called schemes over the ‘field of one element’ $F_1$ of various authors, dg-schemes, slightly noncommutative D-schemes etc. Many ideas of scheme theory and the spectral theory of rings have influenced parallel developments in the analytic setup (Stein manifolds (see the English Wikipedia), rigid analytic spaces, etc.) and in the noncommutative setup give rise to noncommutative algebraic geometry. Deligne has also suggested how to do algebraic geometry in an arbitrary symmetric monoidal category.
Grothendieck took the viewpoint that the schemes, algebraic spaces etc. are sheaves on $Aff$ in some subcanonical Grothendieck topology (functor of points point of view).
Algebraic geometry starts with study of spaces that are locally modeled on (objects in the category) Aff = CRing${}^{op}$ – main categories being of algebraic schemes and of algebraic spaces; one also allows infinitesimal thickenings leading to formal schemes and other ind-objects in schemes. Hakim, Deligne and Gabber extend the setup internally to a symmetric monoidal category, where $Aff$ is replaces by the opposite to the category of monoids in that category; Durov on the other side takes monoids in the category of endofunctors in Set, i.e. monads as the opposite to the local objects in a generalized scheme theory.
This is to be compared to and contrasted with for instance differential geometry, that studies spaces locally modeled on (objects in the category) CartSp – called (smooth) manifolds.
The general formalization of the notion of space and hence, by the general lore of space and quantity, that of sheaf, stack, ∞-stack has originally been crucially inspired by Grothendieck’s work on algebraic geometry and is more recently being greatly revived and further extended by the developments in derived algebraic geometry by people like Bertrand Toen and Jacob Lurie.
In particular in Structured Spaces Lurie presents a general formalism of generalized schemes that encompasses the spaces studied in algebraic geometry and derived algebraic geometry just as well as ordinary smooth manifolds, derived smooth manifold and harmonizes with other axiomatizations such as in synthetic differential geometry: all of these space object are realized as special cases of structured (∞,1)-toposes, differing only in the choice of geometry (for structured (∞,1)-toposes) that they are modeled on.
From this perspective,
ordinary algebraic geometry is the study of structured (∞,1)-toposes for the Zariski or etale geometry $\mathcal{G}_{Zar}$, $\mathcal{G}_{et}$ on CRing${}^{op}$. In fact one has as series of geometries for every integer $n\geq 0$, where classical case is at level $0$ and derived at level $\infty$. Cf. 4.2.9 in Structured Spaces. The fully faithful embedding of schemes into derived schemes does not commute with limits, what is relevant e.g. for intersection theory.
derived algebraic geometry is the study of structured (∞,1)-toposes for the Zariski or etale pre-geometry $\mathcal{T}_{Zar}$, $\mathcal{T}_{et}$ on CRing${}^{op}$.
Despite of this, an axiomatic formulation of algebraic geometry along the lines of synthetic differential geometry, that would de-emphasize the peculiarities of $CRing^{op}$ and emphasize structural aspects such as to facilitate for instance the transportation or interpretation of results in algebraic geometry to other geometries, is currently hardly to be found in the elementary literature. SGA, specially SGA IV was written however to reflect “algebraic” geometry over any topos.
Maybe we could talk more about synthetic differential geometry applied to algebraic geometry to unify perspective of algebraic with differential geometry.
Discussion of fundamental constructions of algebraic geometry from the perspective of the internal logic of the sheaf topos over a scheme (Zariski topos/etale topos) is in
Ingo Blechschmidt, Using the internal language of toposes in algebraic geometry, talk at Toposes at IHES, November 2015 (pdf, recording)
Ingo Blechschmidt, Using the internal language of toposes in algebraic geometry, thesis (pdf)