What does one call the first (or second) factor of an element of a tensor product? For example, if $V,W$ are vector spaces, and $v \in V$, $w \in W$, with $v \otimes w \in V \otimes W$, how would one refer to $v$? First tensor factor?

This issue is similar to what someone faces when dealing with a polynomial expression $$ c_n\alpha^n + c_{n1}\alpha^{n1} + \cdots + c_1\alpha + c_0 $$ where $\alpha$ actually satisfies an equation of degree smaller than $n$. Logically speaking such expressions can be written in multiple ways (consider a quartic polynomial expression in $\sqrt{3}$), but nobody has a problem speaking about the $i$th term in the expression. Just do the same thing when you write down an elementary tensor $v_1 \otimes v_2 \otimes \cdots \otimes v_k$: call $v_i$ the $i$th component (or $i$th term, or perhaps even the $i$th factor). Now comes an issue of who your audience is (which you didn't indicate). If your audience is experts, then it would be clear to your audience that whatever you're doing with $v_i$ is eventually leading to some welldefined result in terms of the tensor itself, so there's nothing more to say. If your audience is students, to whom the tensor product is still somewhat new, then be sure to remind them that mathematically an elementary tensor does not have welldefined components, since an elementary tensor could be written as an elementary tensor in multiple ways. You might then mention the example of polynomial expressions as above which could be written in multiple ways, as an analogy. 

