What is a zero morphism in an abelian category I am trying to familiarize myself with some basic category theory 
and I am getting confused with what a $0$-morphism is.
If we are in category of say $k$-vector spaces
then I am guessing $0$-morphism would be the map that sends everything to $0$.
In these examples it makes more sense to me because each object has
this $0$ element. But it's not really clear to me what happens in more
abstract cases.
I would appreciate if someone could explain to me how I should think
of these $0$-morphisms. Thanks!

Another way to put it.
An abelian category $\mathscr M$ is in particular a category enriched over the category $\mathsf{Ab}$ of abelian groups. So for two objects $A,B$ of $\mathscr M$, the hom-set $\hom_{\mathscr M}(A,B)$ actually carries a structure of abelian group : in particular, it has a neutral, which is the zero map from $A$ to $B$.
