Q:

Correct name for multi-dimensional array/matrix/tensor

What is the correct name for an n-dimensional array in mathematics? I have seen the following:

nD-Matrix
nD-Array
nD-Tensor

Which is the right way?

A:

"Array" is not used much in math, except possibly by some computational math authors. I generally think of that as computer science terminology. I think they also allow for non-number objects to appear as entries (like strings, for example), and some people are uncomfortable calling something with string entries a matrix.
"Array" seems to be used to encompass any sort of list of numbers in one shape or another, usually when you identify an object by numerical components. Vectors, linear transformations and tensors (=multilinear transformations) all have coordinate representations, if you fix a basis.

linear transformations generalize to tensors.
matrices represent linear transformations the way "multidimensional arrays" represent tensors
(Here is an informal diagram that I hope is not too misleading, and which I hope people do not take too seriously.) "vector $\rightarrow$ linear transformation $\rightarrow$ tensor" is analogous to: "coordinates of vector $\rightarrow$ matrix $\rightarrow$ multidimensional matrix"

I'm using "array" in "multidimensional array" because some people have a hard time thinking of a matrix as anything besides a flat object, but you can also say "multidimensional matrix". While matrices have rows and columns, the multidimensional versions can "go in more directions". 
A matrix has componets with two indices. If your array has items with three indices, you could make them into a cube-matrix. If it has four indicies on each entry, you could make it into a hypercube-matrix etc.

