A tensor is an abstract object generalising a scalar or vector and can be represented by a number, a 1D array, 2D matrix or higher order generalisations thereof. A tensor product is a product defined on these tensors yielding other tensors or a method to define or represent tensors. If appropriate, also use the [mathematics] tag.
What are tensors?
A tensors is an $\left(n, m\right)$-order structure that generalizes scalars and vectors to higher orders and can be represented by a number, array, matrix, or higher order generalisations thereof. In physics, tensors with a 3- or 4- dimensions are common, but higher dimensions are sometimes used. For a formal definition, consult the standard sources, e.g. Wikipedia.
What about their product?
An elementary tensor is defined as being non-zero and completely factorisable. Here, the rank of a tensor is the minimum number of elementary tensors that sum to that tensor, meaning that an elementary tensor is 'rank 1'. As such, impure tensors (having a rank greater than 1) are not factorisable, so are hard to represent and hence complicate matters. However, there is a lot of mathematics done on these structures that is ready for Physics to use. A good resource is again Wikipedia.
Why do the quantum physicists care?
Well, one common example of tensors in physics is the stress tensor. But tensors pop up in quantum computation in the definition of quantum gates and other places. As quantum states are vectors, they are tensors, so larger pure states are described using tensor products of the smaller states. When entangled, sums of these states are used, giving a rank greater than 1.
I still don't understand what tensors are. Help?
Personally, as someone with a mathematics background, I don't understand the physicists when they talk tensors. However, I can understand the mathematicians. So, perhaps you can try to get someone different to explain. That can help.