Given a quadratic form Q defined on a vector space V over a field F, the Clifford algebra of Q is an associative F-algebra C with a vector space homomorphism f : V to C such that f(v)2 = Q(v) for all v∈V. Furthermore, the triple (C, V, f) has the universal property that if A is any associative algebra with a homomorphism g : V to A such that g(v)2 = Q(v) for all v∈V, then there is a unique algebra homomorphism h : C to A such that hf = g. It can be shown that f is injective and therefore we may identify V with its image in C. If the dimension of V is n, then the dimension of C is 2n. We shall refer to V as the quadratic space of C.
The primary references for quadratic forms and Clifford algebras are [Che97] and [Art57]. A more recent account, with applications is [Lou01].