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.