What is a Tensor Product?

In linear algebra, we deal with vector spaces and linear maps, and we are happy. But we also want to talk about inner products, bilinear forms and such things, and these are not linear maps. Instead of developing a completelu new theory, we do what we do best: reduce the problem to one we know how to solve. The tensor product allows us to turn bilinear maps $V\times W\to U$ into linear maps $V\otimes W\to U$. And this makes us happy again. This is important, because we like being happy.

Once we have established that, I would explain the actual construction and basic properties of the tensor product to an undergraduate by pointing him to the library. People with a reasonable background of linear algebra are perfectly prepared, in my experience, to read the construction and so on.

https://www.dpmms.cam.ac.uk/~wtg10/tensors3.html

Timothy Gowers posted this on his website. He has a very good explanation of tensor product, it's definition, and why we use it. I found it very useful when I was beginning my study of them.