tensor
A tensor is a single object that packages a multilinear relationship among vectors and measurements — a generalization that contains scalars, vectors, covectors, and matrices as special low-order cases. Intuitively it is a multidimensional array of numbers that transforms in a controlled way under change of basis, but the modern view drops the array and keeps the intrinsic object.
Precisely, a tensor of type (p, q) on a vector space V is an element of the tensor product V^{⊗p} ⊗ (V*)^{⊗q}; p is the contravariant rank (number of V-factors, upper indices) and q the covariant rank (number of V*-factors, lower indices). Equivalently, by the universal property, such a tensor is a multilinear map taking q vectors and p covectors to a scalar. Type (0, 0) tensors are scalars, (1, 0) are vectors, (0, 1) are covectors, (1, 1) are linear maps, and (0, 2) are bilinear forms.
Operations on tensors include the tensor product (raising total rank), contraction (lowering rank by pairing a V-slot with a V*-slot), and — when V carries a nondegenerate bilinear form — raising and lowering indices. In coordinates a type (p, q) tensor is recorded by a multi-indexed array of n^{p+q} components, but the array's meaning depends on the basis; the tensor itself does not.
A linear map A: V -> V is a (1, 1) tensor, recorded as a matrix [a_ij]; its components transform as P A P^{-1} under a basis change by P. A bilinear form is a (0, 2) tensor, recorded as a Gram matrix transforming as P^T G P.
Different tensor types transform by different rules — similarity versus congruence.
The physicist's slogan 'a tensor is something that transforms like a tensor' is the same statement as basis-independence: the abstract element of V^{⊗p} ⊗ (V*)^{⊗q} is fixed, and its component array changes by the corresponding power of the change-of-basis matrix and its inverse-transpose.