Algebraic geometry is the study of the geometric properties of sets of solutions to algebraic equations. Many well-known manifolds can be described in this way (e.g. the -sphere is the set of solutions in to the equation ), and so in principle algebraic geometry has a lot of overlap with other areas of geometry. However, two characteristic features of algebraic geometry are: one very often works over an algebraically closed field, such as (so that parts of algebraic geometry have quite a lot of overlap with the theory of complex manifolds, or complex analytic geometry); one very often considers projective varieties rather than affine varieties. (Thus the -sphere is the set of real points of an affine variety; the set of complex solutions of the associated projective variety is a quadric surface.)
Algebraic geometry can use both geometric methods and algebraic methods. When it does use algebraic methods, it often interprets them in geometric terms. This opens up the possibility of using geometric ideas, and geometric intuition, even when studying algebraic problems over the integers or rational numbers, or even over finite fields. Thus algebraic geometry plays an important role in the study of Diophantine equations.