A post-main-sequence star is a star that has been through a main-sequence phase (the primary, lengthy phase of hydrogen-burning fusion in the center, i.e., it has reached TAMS, terminal-age main sequence) and is proceeding through additional phases, sometimes including fusion of various elements. The phases it goes through, which depend upon the mass of the star, potentially include giant star phases. As in the earlier of the life of the stars, the most massive (i.e., "early") stars proceed through these phases more rapidly.
The lightest stars (red dwarves, on the order of a half solar mass or less) are convective through their main sequence keeping the constituents mixed and won't produce a core that is sufficiently hot and sufficiently dense in helium to result in helium fusion when the hydrogen fusion has ceased. Their main-sequence lifetime is far too long for any to have reached such a point, but stellar modeling suggests they will simply cool down, or will grow hotter during their main sequence (sometimes the term blue dwarf is used) and collapse into a white dwarf.
If a star is more massive than a red dwarf, it passes into giant star phases:
During their AGB phase, stars above roughly 8 solar masses generate carbon (producing additional oxygen plus heavier elements), and depending upon their mass, generate further fusion (carbon burning, oxygen burning, neon burning, and eventually alpha process leading to iron) for the most massive. In these massive stars, these post-main-sequence stages are short in duration (as was their main sequence stage), as short as 1000 years total for the most massive. If massive enough (above roughly 25 solar masses), they eventually undergo a core collapse supernova leaving a neutron star or stellar black hole (one estimate of the main-sequence mass necessary to result in a black hole is 29 solar masses). Above about 40-50 solar masses, they undergo the collapse, but the bright explosion does not occur, i.e., no supernova. The less-massive stars, without the final, drastic "core collapse", produce a planetary nebula, and eventually leave a white dwarf. (All these numbers are approximate at best: subject to further research and depending upon various factors such as metallicity.)