A compact object (or compact star, sometimes abbreviated CO) is a general term for small, very massive stellar objects, typically meant to include white dwarfs, neutron stars, black holes, and any other hypothesized exotic dense stars. They are typically stellar remnants, what's left after a star's fusion has run out. The term degenerate star is also used for compact objects that are not black holes.
A compact object merger is a merger of two such stars, with a luminosity potentially 1015 times that of the Sun, and sometimes a source of detectable gravitational waves.
The term central compact object (CCO) is used for a compact object centered within some observed larger object such as a planetary nebula or supernova remnant.