nLab
finitely generated object

An object $A$ of a concrete category $C$ is finitely generated if it is a quotient object of some free object $F$ in $C$, where $F$ is free on a finite set. (We probably want ‘quotient object’ here to be interpreted in the sense of a regular epimorphism from $F$ to $C$, although perhaps we explicitly want it to be the quotient of a congruence on $F$.)

The object $A$ is finitely presented if it is the quotient of a congruence $R\rightrightarrows F$ such that $R$ is also free on a finite set. (Or maybe we don't demand that this is a congruence but accept any coequaliser of this form?)

See discussion at finitely presentable object more a more abstract version of these.