A vector space (over some field) with an additional binary operation, a vector-valued product between vectors, which is bilinear over vector addition and scalar multiplication. (N.B.: such bilinearity implies distributivity of the vector multiplication with respect to the vector addition, which means that such a vector space is also a ring.)
An algebraic structure studied therein.
A branch of mathematics dealing with equational classes of algebras, where similar theorems from disparate branches of algebra are unified.