In the mathematical field algebra, an algebra is a
vectorspace A over a field F with an
associative product on the
vectorspace - a mappping A x A -> A. The are non
associative algebras out there, too.
There special types of algebras, e.g. group algebras where a base of the vectorspace is a group by the algebra multiplication.
Please note that this definition doesn't directly coincide with the definition of a boolean algebra. A boolean algebra would be here a (Z/2Z) - algebra with the "xor" as vectorspace addition and the "and" as algebra multiplication.
An algebra usually is also a ring.
Further information should be found in any good algebra book.