There are two different notions of dense subcategory $D$ of a given category $C$:
A subcategory $D\subset C$ is dense if every object in $C$ is canonically a colimit of objects in $D$.
This is equivalent to saying that the inclusion functor $D\hookrightarrow C$ is a dense functor.
An older name for a dense subcategory in this sense is an adequate subcategory.
A subcategory $D\subset C$ is dense if every object $c$ of $C$ has a $D$-expansion, that is a morphism $c\to\bar{c}$ of pro-objects in $D$ which is universal (initial) among all morphisms of pro-objects in $D$ with domain $c$.
This second notion is used in shape theory. An alternative name for this is a pro-reflective subcategory, that is a subcategory for which the inclusion has a proadjoint.
there is also the notion of “dense subsite”, but this is not a special case of a dense subcategory.
See the relevant section of MacLane’s Categories for the Working Mathematician.