In mathematics, what is an $N \times N \times N$ matrix?

Tensors in general are multilinear coordinate-free objects that can be represented with respect to some basis by a multi-dimensional arrays indexed appropriately. Just like you can represent a bilinear form $B \colon V \times V \rightarrow \mathbb{F}$ when $V$ is an $n$-dimensional vector space (or a linear map $T \colon V \rightarrow V$) by a $n \times n$ matrix and use a matrix (together with a choice of basis) to define a bilinear form (or a linear map) on $V$, you can represent a multi-linear map $B' \colon V \times V \times V \rightarrow \mathbb{F}$ ("a tensor") by an $n \times n \times n$ dimensional array of scalars and use such an array to define a multi-linear map on $V$.

You can think of a rank three tensor as a three dimensional array. A matrix is a rank two tensor, or a two dimensional array. A vector then is a rank one tensor and scalar a rank zero. This is a simplification of the subject of tensors, but it's useful to think of them as a generalization of matrices.