Tensor: Definition, Examples & n,m Meaning

[subsonicman]

I was reading this page: http://en.wikipedia.org/wiki/Tensor
which said the definition of a tensor was a relation between two vectors. I then went down to the examples section and it had some sort of (n,m) notation. I had some theories on what they meant but none of them made sense. What do n and m represent?

 

[WannabeNewton]

$n$ is just the number of products of $V^{*}$ and $m$ is the number of products of $V$ which comprise the domain of the map; the codomain is just the reals.

 

[HomogenousCow]

which said the definition of a tensor was a relation between two vectors.

That is hardly true, a tensor is a linear object which maps n vectors and m one-forms into real numbers, and transforms in a coordinate invariant manner.
That is like saying multiplication is defined as a relation between two numbers.