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!

The zero morphism $A \to B$ can be factored into
$$ A \to 0 \to B $$
where $0$ is a zero object. (i.e. it is a terminal object and an initial object)
As an aside, when you wrote it as "$0$-morphism", my first reaction was that you were referring to the concept from higher category theory; e.g. in $\mathbf{Cat}$, categories are $0$-morphisms (i.e. objects), functors are $1$-morphisms, and natural transformations are $2$-morphisms.
