I'm familiar with the tensor product of modules, but I've also come across functor tensor product (in emily riehls paper on homotopy limits), what are they, and how are they (if they are) related to traditional tensor products? (Emily shows that they can be defined as a particular coend, but that doesn't really provide any intuition for me). 



What is the functor tensor product?
It is easy to be explicit. Not in full generality, given a (small) closed symmetric monoidal category $\mathcal C$ with coequalizers, a covariant functor $M\colon \mathcal C\to \mathcal C$ and a contravariant functor $N\colon \mathcal C \to \mathcal C$, the tensor product 
$N\otimes\_{\mathcal C} M$ is the coequalizer of the diagram 


$$\coprod\_{(c,d)} N(d) \otimes \mathcal C(c,d) 
\otimes M(c) \implies \coprod\_{e} N(e)\otimes M(e).$$


Here $c,d,e$ range over the objects of $\mathcal C$ and $\implies$ indicates a pair of arrows;
one is given by the evaluation maps $N(d) \otimes \mathcal C(c,d)\longrightarrow N(c)$ of $N$
and the other by the evaluation maps $\mathcal C(c,d)\otimes M(c) \longrightarrow M(d)$ of $M$.
The similarity to Mike's special case should be clear. This is of course an example of a coend,
but I prefer to use the tensor product notation in this special case to make the intuition clear. 


