Endomorphism is such morphism (morphism is another term for homomorphism) whose source and destination are the same object.

That is a morphism $f$ is endomorphism, when $\mathrm{Src}f=\mathrm{Dst}f=A$ where $A$ is some object (e.g. $A$ may be an abstract algebra). Then one can say, the object of endomorphism $f$ is $A$.

In the most general case endomorphisms are encountered in category theory. As a special case of this endomorphisms are also encountered in abstract algebra.

A morphism which is both an endomorphism and an isomorphism is called automorphism.

The sets of endomorphisms and automorphisms for an object $A$ of a category are often denoted correspondingly as $\mathrm{End}(A)$ and $\mathrm{Aut}(A)$ or sometimes as $\mathrm{end}(A)$ and $\mathrm{aut}(A)$.

Endomorphisms also can be considered as objects of category of intermorphisms and (if the set of morphisms of our category is preordered) also of category of pseudomorphisms.