A classification of all the Clifford algebras is given in terms of Kronecker products of the quaternion and dihedral groups. The relationship to spinors in n dimensions is explicitly determined. We show that the real Clifford algebra in Minkowski spacetime is distinct from both the algebra of Dirac matrices and the algebra of Majorana matrices, and cannot be realized by the spinor framework. The matrix representations of Clifford algebras are discussed, and are utilized to give a classification of the real forms of Lie algebras. We are thus able to relate Clifford, Lie, and spinor algebras in an intrinsic geometrical setting.