symmetric algebra
The symmetric algebra Sym(V) is the mirror image of the exterior algebra: instead of forcing antisymmetry, you force commutativity. You build it from the tensor algebra by imposing v*w = w*v for all vectors, so the order of factors no longer matters. Its graded pieces Sym^k V consist of fully symmetric k-tensors — products that are unchanged under permuting the factors.
Its punchline is an identification you already know in disguise: Sym(V) is canonically the algebra of polynomials in coordinates of V. If V has basis x_1, ..., x_n, then Sym(V) is the polynomial ring in x_1, ..., x_n, and Sym^k V is the space of homogeneous polynomials of degree k. So 'symmetric tensor of degree k' and 'homogeneous degree-k polynomial' are the same object.
Dimensions follow the multiset count: dim Sym^k V = (n + k - 1 choose k), the number of degree-k monomials in n variables. Contrast this with the exterior power (n choose k): symmetric tensors keep growing without bound as k increases, because you can repeat variables freely, whereas wedges die past degree n because repetition is forbidden.
Holding the two algebras side by side clarifies a lot. Antisymmetric (exterior) tensors encode oriented volume, determinants, and forms; symmetric tensors encode polynomials, quadratic forms, and moments. The general tensor power decomposes (in characteristic zero) into symmetric and antisymmetric parts plus more exotic pieces — the start of representation theory of the symmetric group.
The symmetric algebra of R^n is the polynomial ring in n variables; degree-k pieces are homogeneous polynomials.
Degree 2 is the everyday face: Sym^2(V*) is exactly the space of quadratic forms / symmetric bilinear forms on V, which you met as symmetric matrices A with q(x) = x^T A x. So 'symmetric matrix' is just a symmetric 2-tensor.