An extremal epimorphism (also called a cover) in a category $C$ is an epimorphism $e$ such that if $e = m g$ where $m$ is a monomorphism, then $m$ is an isomorphism.
The dual notion is an extremal monomorphism.
If $C$ has equalizers, then any morphism with the property above must automatically be an epimorphism.
Any strong epimorphism is extremal. The converse is true if $C$ has pullbacks.
Any regular epimorphism is strong, and hence extremal. The converse is true if $C$ is regular.
An image factorization of a morphism $f$ is, by definition, a factorization $f= m e$ where $m$ is a monomorphism and $e$ is an extremal epimorphism.