Algebraic geometry is the study of geometries that come from algebra, in particular, from rings.
In classical algebraic geometry, the
algebra is the ring of polynomials,
and the geometry is the set
of zeros of polynomials,
called an algebraic variety. For instance, the
unit circle is the set of
zeros of the polynomial 
and is an algebraic
variety, as are all of the conic sections (Hartshorne
1977, Harris 1992, Shafarevich 2013).
The basic ideas of classical algebraic geometry can be applied to any commutative ring with a unit, such as the integers. The geometry of such a ring is determined by its algebraic structure, in particular its prime ideals. Grothendieck (1960) defined schemes as the basic geometric objects, which have the same relationship to the geometry of a ring as a manifold to a coordinate chart. Eilenberg and MacLane (1945) introduced category theory to study natural equivalences in algebra and topology. Its language subsequently became central to algebraic geometry.
Algebraic geometry also studies how algebraic varieties meet. For example, when two subvarieties of complementary dimensions in a projective space meet in finitely many points, the points can be counted with multiplicity; the total is an intersection number. A point where two algebraic curves are tangent is counted more than once (Fulton 1998, Shafarevich 2013).
As a consequence, algebraic geometry became very useful in other areas of mathematics, most notably in algebraic number theory. For instance, Deligne (1974) used it to prove a variant of the Riemann hypothesis. Also, Wiles' (1995) proof of Fermat's last theorem used the tools developed in algebraic geometry.
In the latter part of the twentieth century, researchers have tried to extend the relationship between algebra and geometry to arbitrary noncommutative rings. The study of geometries associated to noncommutative rings is called noncommutative geometry.
