Motivation for the Tensor Product

In general, if $F$ is a field and $V$ is a vector space over $F$, the tensor product answers the question "How can I define scalar multiplication on $V$ by some larger field which contains $F$?" (Of course this holds if we replace the word "field" by "ring" and consider the same scenario with modules.)

Concrete example: Suppose $V$ is the set of all $2\times 2$ matrices with entries in $F=\mathbb{R}$. In this case we know what "$F$-scalar multiplication" means: if $M\in V$ is a matrix and $c\in \mathbb{R}$, then the new matrix $cM$ makes perfect sense. But what if we want to multiply $M$ by complex scalars too? How can we make sense of something like $(3+4i)M$? That's precisely what the tensor product is for! We need to create a set of elements of the form $$\text{(complex number) "times" (matrix)}$$ so that the mathematics still makes sense. With a little massaging, $\text{"times"}$ will become $\otimes$ and this set of elements will turn out to be $\mathbb{C}\otimes_{\mathbb{R}}V$.

Extending the idea further, we can also construct the tensor product between two vector spaces (and more generally, between two modules). And it's this construction which answers the more general question, "How can I define multiplication between two vectors?"