This is one of my most important RPG because of it combining dark humour with quite nice mechanics and making player conflict a pleasurable part of the game, rather than a serious problem. I like the setting in Alpha Complex, which is a huge underground bunker ruled by an insane computer and populated by the remains of humanity. In Alpha complex everyone is a mutant, which is illegal and a member of a secret society, also illegal. Characters are troubleshooters recruited by the computer to perform missions including seeking out members of secret societies or mutants and eliminating them. Troubleshooters missions can involve exciting chances to try new R&D equipment with its famous reliability or helping their friend the computer in many other ways.
The rules are simple and fairly easy to understand, though as a GM there was always the nice touch that any player knowing too much on the rules was obviously a traitor and therefore their current clone could be terminated preferably in an amusing way (amusing to the GM that is). The characters have a skill system which is used against a D20 roll with the aim being to roll below the skill level, this gives a clean easily understood definition of success though of course the GM should make sure as many potentially random modifiers as possible are applied so that players are not always sure what will be successful. There are no classes or levels in the game though within society clones have a security clearance between the Infrared drones at the bottom through to the powerful Ultraviolet High Programmers at the top. In a secret society a character can have a rank with this influencing what benefits they can get from their membership.
There is possibly more scope for different types of games with Paranoia than with many supposedly straight RPG as it can be played in the classic manner with a darkly humorous approach, as an outright straight RPG of a grim future or with the comedy element increased. Personally I've always favoured the classic manner as I like the combination of humour with the darker subject matter.
Paranoia is an unusual RPG in that player cooperation is deliberately made limited with all characters having good reasons to turn in the others as traitors and make themselves more popular with the computer. It is also nice in that it makes rules lawyering pretty much impossible, as arguing with the GM about the rules is one of the quickest ways to get a character terminated (at least till you run out of clones). Long term character development is again a small part of the game as it is rare that a player will not run out of clones before any character has completed a few missions with pregenerated characters often being a better way to play the game.
I have the hardback Games Workshop reprint of the rules which incorporated several small adventures and a number of others I bought at the time and recently on e-bay. Paranoia was one of the few RPG that kept my interest in the twilight years of my gaming between the late '80s and the start of the new century precisely because of the simplicity of the rules and the break with the standard tropes of RPG.
In short: Paranoia is fun, trust the computer, the computer is your friend. Other games are not fun and playing them is treason.
I was glad to see this revived by Mongoose with a new edition written by Allen Varney in 2004. To date I've not picked up the new Paranoia, but it has been well reviewed at RPG.net