Wikipedia defines a Clifford algebra as a special sort of Associative algebra which is in turn a special sort of Vector space. Their definition of associative algebra says specifically that products of vectors are vectors. (A x A ➝ A). This must be so in Clifford algebras as well, but some developments of Clifford algebras say of basis elements b, that bb = −1. How can that be? −1 is a scalar, not a vector!
The Wikipedia development describes a process starting from an n-dimensional vector space V over a field K and a quadratic form Q for V. The result is a new vector space C of 2n dimensions. There are within C two subspaces, one naturally isomorphic to V and another naturally isomorphic to K. Much of Clifford algebra literature identifies both V and K with their natural counterparts in C. This risks notation confusion: is αu the vector space multiplication of a scalar by a vector, or is it the new algebraic multiplication between two elements of C? Another confusion is whether αβ is a field multiply, a scalar by vector multiply, or the multiply defined in the new algebra. In short is αβ a construct in K, V or C?
You might have guessed that by the axioms the three meanings are all the same. I have never seen notation introduced in which you could prove that these syntactically ambiguous expression are not semantically ambiguous. It is reminiscent of the reals being honorary members of the complex. It is why these constructs are called hypercomplex numbers.
There are several developments on the web and the Wikipedia is the most abstract, requiring more preliminaries, but is also coordinate free. I will follow the more pedestrian and more common development here.
Choose a basis γi for V, 0≤i<n.
New Axioms: (1) if i≠j then γiγj = −γjγi,
(2) γiγi = −1.
The axioms from associative algebras now define the product of any two vectors when we know the products of any two base vectors. The γs are base vectors for V, not C. What are base vectors for C? We cannot prove from the axioms that γ1γ2 ≠ γ3 but we seek the largest algebra generated by K and so we assume that the products of base elements of K are distinct except as required by the axioms. Any product of base elements can be rewritten in increasing order in the subscripts. That will cause some sign reversals. There can be no repeated factors in this expression since γiγi = −1. In short for each of the 2n subsets of the set of n base vectors of V, there is exactly one base vector for C, those base elements from base vectors of V multiplied together. This is a complete base set for C. It also proves that C has 2n dimensions.
Another was to define Cn, the Clifford algebra over Vn is to define C0 as the reals and to define Cn as an algebra of pairs: <u, v> where u, v ∊ Cn−1. The rest of this is developed here.