multilinear map
A multilinear map takes several vectors as input and is linear in each slot separately, holding the others fixed. For two slots this is a bilinear map B(v, w); for k slots it is k-linear: B(..., a*x + b*y, ...) = a*B(..., x, ...) + b*B(..., y, ...) in every position. Fix all but one argument and you get an ordinary linear map; that is the defining feature.
Examples are everywhere. The inner product <u, v> is bilinear (over the reals). The determinant is n-linear in the columns of an n-by-n matrix. The cross product is bilinear. A bilinear form is a multilinear map whose output is a scalar; a multilinear map can also output vectors, like multiplication in an algebra.
The decisive fact is that multilinear maps are NOT linear on the product space. B(v, w) is not linear in the pair (v, w) jointly: B(2v, 2w) = 4*B(v, w), not 2*B(v, w). So you cannot treat them with ordinary linear-algebra machinery as they stand — which is exactly the problem the tensor product was invented to fix.
That is the headline relationship: tensors are the universal recipients of multilinear maps. Every k-linear map on V1 x ... x Vk factors uniquely through a single linear map on V1 (x) ... (x) Vk. So 'multilinear' and 'tensor' are two views of one idea — the input side and the linearized side of the same machine.
The determinant is the prototypical multilinear (and alternating) map in the columns.
Watch the difference between multilinear and linear on the product: a linear map L(v, w) on V (+) W satisfies L(2v, 2w) = 2*L(v, w), but a bilinear B scales the inputs together as a product, giving B(2v, 2w) = 4*B(v, w). Same arguments, totally different rule.