[1] That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
The prefix super- comes from the theory of supersymmetry in theoretical physics.
Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry.
The study of such objects is sometimes called super linear algebra.
Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, supermanifolds and superschemes.
In most applications, K is a field of characteristic 0, such as R or C. A superalgebra over K is a K-module A with a direct sum decomposition together with a bilinear multiplication A × A → A such that where the subscripts are read modulo 2, i.e. they are thought of as elements of Z2.
For this reason, commutative superalgebras are often called supercommutative in order to avoid confusion.
[2] When the Z2 grading arises as a "rollup" of a Z- or N-graded algebra into even and odd components, then two distinct (but essentially equivalent) sign conventions can be found in the literature.
This is more often seen in physics texts, and requires a parity functor to be judiciously employed to track isomorphisms.
Detailed arguments are provided by Pierre Deligne[3] Let A be a superalgebra over a commutative ring K. The submodule A0, consisting of all even elements, is closed under multiplication and contains the identity of A and therefore forms a subalgebra of A, naturally called the even subalgebra.
If A has no 2-torsion (in particular, if 2 is invertible) then the grade involution can be used to distinguish the even and odd parts of A: The supercommutator on A is the binary operator given by on homogeneous elements, extended to all of A by linearity.
The graded tensor product of two superalgebras A and B may be regarded as a superalgebra A ⊗ B with a multiplication rule determined by: If either A or B is purely even, this is equivalent to the ordinary ungraded tensor product (except that the result is graded).
The definition given above is then a specialization to the case where the base ring is purely even.
A superalgebra over R is a R-supermodule A with a R-bilinear multiplication A × A → A that respects the grading.
Bilinearity here means that for all homogeneous elements r ∈ R and x, y ∈ A. Equivalently, one may define a superalgebra over R as a superring A together with an superring homomorphism R → A whose image lies in the supercenter of A.
An associative, unital superalgebra over R can then be defined as a monoid in the category of R-supermodules.
That is, a superalgebra is an R-supermodule A with two (even) morphisms for which the usual diagrams commute.