Subspaces of a tensor product of vector spaces

Clearly if $V_1,W_1$ are subspaces of $V,W$ respectively then $V_1\otimes W_1$ is a subspace of $V\otimes W$. But these are not all vector subspaces of $V\otimes W$.

For instance take $V_2 < V$ and $W_2<W$ such that $V_1\cap V_2=\{0\}$ and $W_1\cap W_2=\{0\}$; then $(V_1\otimes W_1)\oplus (V_2\otimes W_2)$ is also a vector subspace of $V\otimes W$, but it cannot be expressed in the form above.

Well, do you know how to describe "the subspaces" of a generic vector space? In general, vector spaces are easy to describe (their isomorphism class, anyway): they are all $\cong F^n$ where $F$ is the scalar field and $n$ is the dimension. (Although strange things happen without the axiom of choice.)

So if someone asked "what are the subspaces of $F^n$?" what would you say to them?

Note that $\dim(V\otimes W)=\dim(V)\times\dim(W)$ so what $V\otimes W$ as a vector space is completely determined by the dimensions of $V$ and $W$. In general, any vector space looks like the tensor product of two spaces: indeed $V\cong V\otimes_FF\cong F\otimes_FV$ for any vector space $V$ over $F$. So your question really does reduce to asking what the subspaces of an arbitrary space are.

It's not the case that subspaces of $V\otimes W$ look like $V_1\otimes W_1$. Indeed, if $V=W$, then the tensor square $V^{\otimes n}=V\otimes V$ has the subspace ${\rm Sym}^2(V)$ of symmetric tensors and ${\rm Alt}^2(V)$ of alternating tensors, and neither of these look like $V_1\otimes W_1$. Observe that when forming a pure tensor $v\otimes w$ in a subspace $V_1\otimes W_1$, each of $v\in V_1$, $w\in W_1$ can be chosen independently, but in general with any given subspace $U\subset V\otimes W$, the two factors of a pure tensor might not be so independent. In the case of the ${\rm Sym}^2(V)$ the two factors must the same, and in the case of ${\rm Alt}^2(V)$, there is still some freedom but the two factors must at least be different.