A Deathday Party is a celebration thrown by ghosts to commemorate the anniversary of their death. What seems to be a typical Deathday Party entails a candlelit dinner of rancid, mouldy food, the smell of which would cause a normal living creature to gag. The reason for this is apparently the claim that a ghost can "almost taste" seriously spoiled food.
Deathday Parties seem to the living to be rather gloomy affairs. Harry Potter, Ron Weasley, and Hermione Granger attended Sir Nicholas de Mimsy Porpington's 500th Deathday Party in 1992. Sir Nicolas was deeply touched that the three students put in an appearance.
The only people normally invited to a Deathday Party are no longer living, as noted on the occasion above by Hermione Granger.