What academic functions does a Professor Emeritus usually serve?

To the best of my understanding, the primary function of a professor switching to emeritus status is that it frees up a faculty slot for a new hire. Emeritus is essentially retirement without giving up affiliation. An emeritus professor can ramp down their duties, go part time, etc. In some cases they may still do some teaching and supervising, and may have office space, but I believe they are typically no longer paid and no longer expected to fulfill normal faculty duties. That said, when they are still active, their advice and participation is often still quite valued by the active faculty.

Emeritus faculty are a subset of retired faculty. At some universities, one gets the title "professor emeritus" practically automatically when one retires; at others, the title is a non-trivial honor. (Dean Rusk is reported to have explained "emeritus", at the time he retired, with the etymology: "e" means "out", and "meritus" means "deserves to be".)