Let $C$ be a locally small category that admits filtered colimits of monomorphisms. Then an object $X \in C$ is finitely generated if the corepresentable functor
preserves these filtered colimits of monomorphisms. This means that for every filtered category $D$ and every functor $F : D \to C$ such that $F(f)$ is a monomorphism for each morphism $f$ of $D$, the canonical morphism
is an isomorphism.
An object $A$ of a concrete category $C$ is finitely generated if it is a quotient object (in the sense of a regular epimorphism) of some free object $F$ in $C$, where $F$ is free on a finite set.
The object $A$ is finitely presented if it is the coequalizer of a parallel pair $R \rightrightarrows F$ such that $R$ is also free on a finite set.
A set $X$ is a finitely generated object in Set iff it is (Kuratowski-)finite. For this to hold constructively, filtered categories (appearing in the definition of filtered colimit) have to be understood as categories admitting cocones of every Bishop-finite diagram.
For $R$ a ring, an $R$-module $N$ is a finitely generated module if it is a quotient of a free module with a finite basis.
The general definition is in Locally Presentable and Accessible Categories, definition 1.67.