Trace of tensor product I am currently struggeling with the following problem: 

Imagine that you have 4 vectors $v_1,v_2,v_3,v_4$ in $\mathbb{R}^2$ and then you want to calculate $\hbox{trace}(v_4 \otimes v_3 \cdot  v_1^T \otimes v_2^T)$. What is this? 

I came across this thing in Theoretical Physics dealing with density matrices, but I do not know how to calculate this.

This is a variation of physicist Dirac, bra-ket notation. In a more standard bra-ket notation your equation reads $$ trace ( |v_4 \rangle \langle v_1 | \otimes |v_3 \rangle \langle v_2 | ) $$
Using first $ trace ( A \otimes B) = trace (A) trace (B)$, this evaluates nicely to  (simply "permute cyclically the kets to the left") 
$$ \langle v_1 | v_4 \rangle \langle v_3 |v_2  \rangle \, $$
This is why this notation is indeed useful. By the way, the above means
$$ v_1^T v_4 \,\, v_3^T v_2 \,. $$
In bra-ket notation $|v\rangle$ denotes a vector and $\langle v|$ its adjoint under the scalar product you are using. In your case, it seems you are using real field and (standard) real scalar product
$\langle x|y\rangle = x^T y $
