e.g. if I tell you I have a matrix M and under change of basis it transforms as A^t M A, then I know it's representing a bilinear map of the form V⊗V→F. etc.
Similarly, if I tell you what kind of multilinear "thing" a tensor T is representing, then that tells you how it transforms under change of basis, and vice versa. For 3-tensors, there are several natural possibilities (up to permuting indices):
U⊗V⊗W→F
U⊗U⊗V→F
U⊗U⊗U→F (trilinear map)
U⊗V→W (bilinear map)
U⊗U→V (bilinear map)
U⊗V→U (linear action of V on U)
U⊗U→U (algebra, not nec. associative)
U→V⊗W
U→U⊗V (coaction)
U→U⊗U (coalgebra, not nec. coassociative)
F→U⊗V⊗W
F→U⊗U⊗V
F→U⊗U⊗U