The notion of morphism in category theory is an abstraction of the notion of homomorphism.
In a general category, a morphism is an arrow between two objects.
Given two objects in a (locally small) category, say $x$ and $y$, there is a set $hom(x,y)$, called a hom-set, whose elements are morphisms from $x$ to $y$. Given a morphism $f$ in this hom-set, we write $f:x \to y$ to indicate that it goes from $x$ to $y$.
More generally, a morphism is what goes between objects in any n-category.
The most familiar example is the category Set, where the objects are sets and the morphisms are functions. Here if $x$ and $y$ are sets, a morphism $f: x \to y$ is a function from $x$ to $y$.
morphism, mutlimorphism?