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 
and is an algebraic variety, as are all of the
conic sections (Hartshorne 1977, Harris 1992, Shafarevich
2013).
In the twentieth century, it was discovered that 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 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. The language of category theory evolved at around the same time, largely in response to the needs of the increasing abstraction in 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 used it to prove a variant of the Riemann hypothesis. Also, Andrew Wiles' 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.
