the exterior algebra
Start with an ordinary vector space V — say R^3 with its arrows. The exterior algebra is a way of multiplying those arrows so that the product remembers an oriented area or volume rather than a single number. Two arrows multiply to give a little oriented parallelogram; three arrows give an oriented box. The crucial rule is that swapping two factors flips the sign, so a vector times itself is zero: there is no area in a degenerate parallelogram squashed onto a line.
Formally, the exterior algebra Lambda(V) is the tensor algebra of V quotiented by the relations v ^ v = 0 for every v, which forces v ^ w = - w ^ v. It splits as a direct sum Lambda^0(V) + Lambda^1(V) + ... + Lambda^n(V) where Lambda^k(V) consists of degree-k elements (k-vectors), n = dim V, and Lambda^k has dimension C(n, k) = n! / (k! (n-k)!). If e_1, ..., e_n is a basis of V, then the products e_{i_1} ^ ... ^ e_{i_k} with i_1 < i_2 < ... < i_k form a basis of Lambda^k(V). The top piece Lambda^n(V) is one-dimensional and measures signed n-volume.
This algebra is the home of everything in this field. A differential form at a point is an element of the exterior algebra of the cotangent space T*_p M, the wedge product of forms is this multiplication done pointwise, and orientation is a choice of nonzero element of the top exterior power. A common confusion: Lambda(V) is NOT commutative and NOT anticommutative in a blanket way — the sign of swapping a degree-p factor past a degree-q factor is (-1)^{pq}, so even-degree elements commute with everything. Saying 'the wedge anticommutes' is only literally true on 1-forms.
In R^3 with basis e_1, e_2, e_3: Lambda^0 = R (scalars), Lambda^1 = span{e_1, e_2, e_3} (vectors), Lambda^2 = span{e_1^e_2, e_1^e_3, e_2^e_3} (oriented areas, three of them), Lambda^3 = span{e_1^e_2^e_3} (oriented volume, one of it). Dimensions 1, 3, 3, 1 — the row of Pascal's triangle. And (2 e_1 + e_2) ^ (e_1 - 3 e_2) = -6 e_1^e_2 + e_2^e_1 = -7 e_1^e_2, whose coefficient -7 is exactly the 2x2 determinant of the two vectors.
The wedge of two plane vectors recovers their signed area (a 2x2 determinant); this is why exterior algebra and determinants are the same idea.
Lambda^k(V) and the space of alternating k-linear forms on V are naturally identified only after you also identify V with V* (or work with the dual exterior algebra Lambda^k(V*)). Differential forms live in Lambda^k(T*_p M), the alternating multilinear functionals on tangent vectors — be careful which copy you mean.