Tensor product and Brauer Group

For a project I'm doing, I've been briefed to describe the Brauer Group. I know it has something to do with tensor products of algebras, but my problem is with understanding the tensor product. The textbook I was instructed to work from doesn't even explain what a tensor product of vector spaces is, it's just assumed knowledge. I've been through the first 3 pages of google (literally every link) trying to understand what the tensor product of vector spaces is, but I just can't understand it. I'm looking for a resource that explains tensor products to someone with a background in just linear algebra. If anyone could refer me to a decent resource (or give an explanation), it would be well appreciated! Thanks!

If you take tensor products of finite-dimensional vector spaces over their ground field (i.e. if $V,W$ are vector spaces over the field $k$ and you consider $V \otimes_k W$), then the tensor product is very simply described ; if $V = \langle e_1,\cdots,e_n \rangle_k$ and $W = \langle f_1,\cdots,f_m \rangle_k$ (where the generating sets above are bases for the respective vector spaces), then $V \otimes W$ is the vector space with basis $\{ e_i \otimes f_j \}$. You are also given an extra multiplication map $\otimes$ which allows you to multiply elements of $V$ with elements of $W$ to obtain elements of $V \otimes W$ in a $k$-bilinear way, i.e. $k$-linear in the $V$-argument and in the $W$-argument. The choice of the bases is not relevant ; given different bases, the resulting vector space and $\otimes$-multiplication gives a canonical vector space isomorphism between the two constructions with different bases choice.