TITLE: What are the Differences Between a Matrix and a Tensor?
QUESTION [220 upvotes]: What is the difference between a matrix and a tensor? Or, what makes a tensor, a tensor? I know that a matrix is a table of values, right? But, a tensor?

REPLY [2 votes]: "Tensors are generalizations of scalars (that have no indices), vectors (that have exactly one index), and matrices (that have exactly two indices) to an arbitrary number of indices. " [1]
You can't index into a scalar $s\in \mathbb{R}$. You can index into a vector $v\in \mathbb{R}^d$ with a single index $i$ as $v_i$. You can index into a matrix $M\in \mathbb{R}^{n \times m}$ with two indices $(i, j)$ as $M_{ij}$. 
If I continue adding indices, you might imagine a matrix-like object $T\in\mathbb{R}^{n\times m \times p}$ you can index into with three indices $(i, j, k)$ as $T_{ijk}$. How about $4, 5, 6, ...$ indices? A tensor is such an object that allows any number of indices you want. 
In this way scalars, vectors and matrices are the special case of tensors with 0, 1 and 2 indices. 
[1] http://mathworld.wolfram.com/Tensor.html