Waste extraction
From Memory Alpha, the free Star Trek reference.
The toilet (referred to as waste extraction in the 24th century), was a system intended for the disposal of bodily wastes. The waste extraction system was the system used to manage and process that and all other related wastes.
During the first mission of the starship Enterprise (NX-01), Captain Jonathan Archer declared that the Vulcan High Command didn't think humans could flush a toilet without a Vulcan to assist. (ENT: "Broken Bow")
In 2151, Worley Elementary School in Kenmare, County Kerry, Ireland on Earth, sent questions, letters, and drawings to the Enterprise. One question asked was "When you flush the toilet, where does it go?" and one of the few that was answered personally — Captain Archer delegated it to chief engineer Charles Tucker. Commander Tucker was not pleased with the captain's selection of questions for him and protested, but ultimately relented and explained to Molly the ship's processes of recycling and biomatter resequencing. (ENT: "Breaking the Ice")
The brig aboard Starfleet ships included facilities such as a sink and toilet, which were enclosed behind the wall until needed. A sign above the toilet read "Do not use while in spacedock." (Star Trek V: The Final Frontier)
The microbrain from Velara III caused several malfunctions all over the USS Enterprise-D including locking two crewmembers in the programmers' restroom. (TNG: "Home Soil")
Waste extraction points on a Bajoran transport are located in individual quarters. After Ibudan was murdered in 2369, Dr. Julian Bashir located seofurane fragments near the waste extraction point in Ibudan's quarters on the transport. (DS9: "A Man Alone")
The Ferengi shuttle, Quark's Treasure was equipped with a waste extraction unit. (DS9: "Little Green Men")
On Ferenginar, Quark and Rom's Uncle Pax made a fortune in the waste extraction business; consequently he was also known as "Uncle Stinky". It was said that in the Nagal Residence, the waste extraction fixtures were plated with latinum. Quark, believing that he was about to become the next Grand Nagus, said that the first thing he was going to rip out the plated fixtures and replace them with solid latinum fixtures. (DS9: "Body Parts", "The Dogs of War")
After Enrique Muñiz called Miles O'Brien a "mountain man... an old mountain man", O'Brien asked him how he would like a transfer to Deep Space 9's waste extraction unit. (DS9: "The Ship")
Quark once quipped to Worf that Klingon culture had ceremonies for everything except waste extraction. (DS9: "Looking for par'Mach in All the Wrong Places")
His upcoming wedding with Leeta and preparations for a Dominion attack on Deep Space 9 in late 2373 made Rom so nervous that he announced he had to go to waste extraction. (DS9: "Call to Arms")
In 2375, the USS Voyager faced a situation where it was functioning on a mere four lavatories. (VOY: "Bride of Chaotica!")
On Stardate 48892.1, the Doctor while in a malfunctioning holodeck program (he believed to be real life) requested a ship-wide system update and the computer reported, Sewage and waste reclamation as one of them. At the time, it was presumably offline as with almost all other systems. (VOY: "Projections")
[edit] Background
According to the "Star Trek: The Next Generation USS Enterprise NCC-1701-D Blueprints", the door in Jean-Luc Picard's ready room lead to a bathroom. The blueprints also referred to the bathrooms as both "toilets" and "heads". The main bridge of the USS Enterprise-D featured a bathroom just off the doors leading to the observation lounge. According to Jonathan Frakes during the "Journey's End - The Saga of Star Trek: The Next Generation" special in 1994, the Enterprise-D had only one bathroom and he proceeded to point to it on the large cross-section of the Enterprise in the Main Engineering set.
As for how it works, as Bashir said in "Fascination", best not to dwell on it too much.
