A ceremony to summon Death, for various reasons. The most common cause for wizards to reluctantly summon Death is to ask him questions about inexplicable phenomena, because he seems to know everything. The Rite needs to be performed by eight powerful wizards, either with loads of magical paraphernalia for the look of the thing, or with only a few small bits of essential equipment. Death doesn't like the Rite of AshkEnte, because it so happens that he's always busy when the wizards summon him. Powerful (read: old) wizards dislike the ritual as it makes Death more aware of them, period. Anyone filling the office of Death is called by the Rite, resulting in Susan Sto Helit and Mort having been called once (in Soul Music and Mort.)
It is noteworthy that Capt. Carrot Ironfoundersson has found an easier way to talk to Death that doesn't even need two cc of mouse blood. Maybe it's best for the peace of mind of wizards that they haven't worked this out yet, but Carrot realised that all you need to do is to keep vigil at the bed of a very old man with a terminal illness. When the inevitable happens, you simply detain Death by asking Him to assist the Watch with their inquiries. (As Death is the witness to every murder and suspicious passing, he had a lot to say to Carrot on the subject, for eg, of a sidestall proprietor discovered to have choked to death on his own swizzle) After all, IT'S A FAIR COP. (Refer to Theatre of Cruelty).
Being able to see and converse with Death may, perhaps, be one of those royal prerogatives, that only the true king may enjoy. Or, as seen in Soul Music, where Nobby Nobbs and Fred Colon, who very clearly are not of the blood royal, yet still witness can speak to Death in the small hours of the morning, this is down to the heightened sensitivities of long-time Watchmen and those who frequently see the process of death happening in the small and late hours of the night.
Death really hates it when the Rite is invoked, as the wizards always seem to summon him at the most inconvenient moments. (ie, just before that moment at the masked dance when everyone is due to un-mask at midnight and they think he's going to take his mask off like everyone else). Susan could echo this, as during a moment of intimate conversation with Imp y Celyn when she is just about to tell him what he really,really needs to know, Ridcully decides to give the High Energy Magic wizards a practical lesson in real magic, and performs the Rite.