objects such that commutes with certain colimits
Let be a poset such that every directed subset of has a join; that is, is a dcpo. A compact element, or finite element, of is a compact object in regarded as a thin category; that is, homs out of it commute with these directed joins.
In other words, is compact precisely if for every directed subset of we have
(c \leq \bigvee_i d_i ) \Leftrightarrow \exists_i (c \leq d_i) \,.
Of course, the part of this is automatic, so the real condition is the part. In more elementary terms: