nLab
finitely presentable object

An object being finitely presentable or of finite presentation is the same as it being a compact object. See there for more details.

There is nothing there about the finitely presented objects that I know from algebra, such as finitely presented groups. What is the connection? —Toby Bartels

It does say, under the “In Grp” subsection, that a finitely presented group is the same as a compact object in the category of groups, but I don’t know why that’s true. —Owen Biesel

Mike Shulman: If $C$ is concrete and its forgetful functor to $\mathrm{Set}$ preserves filtered colimits (is “finitary”), then I believe that an object of $C$ is finitely presentable in this sense iff it can be finitely presented in this sense. Possibly there is more generality in which this is true.

There is also an abstract notion of “finitely generated object” meaning that $\mathrm{hom}(X,-)$ preserves directed colimits of monomorphisms, which in the finitary-over-$\mathrm{Set}$ case I think should be equivalent to the more concrete version.

The term “finitely presentable” or “finitely-presentable” is used for instance in

• Kashiwara, Schapira, Categories and Sheaves, Definition 6.3.2;
• Peter Johnstone, Stone Spaces, Definition VI.1.8.