In general, algebra refers to abstract arithmetic, i.e., the general study of arithmetical operations on sets of numbers. In everyday mathematics this typically involves efficiently manipulating expressions or equations involving arthmetical operations, often to determine a value or set of values for an unknown quantity.

In abstract algebra an algebra is a vector space over a set of scalars that satisfies left- and right-distributivity of vector product and whose vector operations commute with the scalars.

The term algebra may be used more loosely to refer to any set with a structure in which operations on the elements may be defined.