"Prendre congé" means "to leave an assembly, a reception, etc." , usually with some words of greeting or thanks, but not necessarily.
"Prendre un congé" is official, and granted for holidays, maternity, studies, etc., for a fixed delay.
It is the origin of two funny expressions.
At the Spanish Court, in the XVIth century, you had to salute every single person of an assembly when leaving.
The French thought it was somewhat too formal and boring to just pay some respects for the hostess, for instance, and the Spanish invented the "Disperdirse a la francesa", promptly translated in English by "Take a French leave", without any tint of real reproach, because in fact everybody thought it to be reasonable.
The French then understood wrongly this, coming from the Perfid Albion, as "nicking off", and took revenge in translating back by "filer à l'anglaise", which is always pejorative.