Let $k$ be a field and write $C_k$ for the category of commutative, associative, unital algebras over $k$. Let us say that an object $A\in C_k$ is finitely presentable if the representable functor $\hom(A,-):C_k\to \mathbf{Set}$ preserves directed colimits.
Is a finitely presentable algebra the same as a finitely generated algebra in the ordinary sense of abstract algebra?
Yes, but this is a somewhat special result. Generally you should expect finitely presentable objects to be the algebraic gadgets with a finite presentation. In a generally algebraic category. having finitely many generators doesn't imply having a finite presentation. For instance, the ideal $(X_1,X_2,...)$ in the polynomial ring in countably many variables is not finitely generated, so the quotient by it is not finitely presented. This doesn't occur over a field, or over a noetherian ring, as ideals of the polynomial rings in finitely many variables are all finitely generated.
