The back of the Paranoia rulebook describes the game as follows: 

Imagine a world designed by Kafka, Stalin, Orwell, Huxley, Sartre and the Marx Brothers...

Paranoia is a roleplaying game set in a darkly humorous future. In Paranoia, a well-meaning but deranged computer desperately protects the citizens of an underground warren from all sorts of real and imagined enemies. 

You play one of The Computer's elite agents. Your job is to search out and destroy the enemies of The Computer. Your worst fear is that The Computer will discover that you are one of these enemies. 

Paranoia: A lighthearted game of terror, death, bureaucracies, mad scientists, mutants, dangerous weapons, and insane robots, which encourages players to lie, to cheat, and backstab each other at every turn. 

Is that fun? 

Trust us.

Alpha Complex

This is where everybody lives. It is a vast self-contained community. Some Games Masters set Alpha Complex underground. Some put it above ground with a big dome over the top. Either way, its occupants have never seen Outdoors. Knowledge of Outdoors is treasonous.

The Computer

The Computer runs everything within Alpha Complex. The Computer sees everything within Alpha Complex (well, that's what It would have you believe). Everybody within Alpha Complex works for The Computer, serves The Computer, protects The Computer and quite often dies for The Computer. (Well, it's sometimes said they often die because of The Computer, but to say such a thing is treasonous. So it's not said very often.) 

Though deranged, The Computer believes it is doing the right thing. It strongly believes that Alpha Complex is in danger of being invaded by Communists or Mutants (or even mutant commies), and to this end it invented Troubleshooter Teams. More about these later.

Clones

The Computer decided that sex and birth were very inefficient methods of creating citizens. Instead, Alpha Complex produces all of its loyal citizens in clone vats. Six citizens are produced at once, all identical. When If a citizen dies, their next clone is sent by means of a rapid replacement system. In practice, this normally means that a replacement clone turns up within five minutes of The Computer being notified of the death of the previous one. 

Despite any evidence to the contrary, everybody believes that a replacement clone is free of any deformities that the previous one may have had (be this a mutation, secret society membership or knowledge of what's really going on). To point out to The Computer that two clones from the same batch have mutations is a bad idea unless you have very good evidence to back this claim up. 

The Computer doesn't make that kind of mistake.

Mutations

Even though modern cloning techniques are perfect (The Computer says so), occasionally a mutation has been known to creep into the genetic material. Although mutations are treasonous, The Computer has been known to overlook mutations if they are reported quickly. From time to time you may see a citizen with a yellow stripe on their uniform. This means they are a Registered Mutant and they have already confessed their mutation to The Computer. 

Details of specific mutations are not available at your security clearance. Should you discover that you have a mutation, or you know of an unregistered mutant, you should report it immediately to The Computer. 

If you should hear rumours that mutations are commonplace, you should report the citizen spreading the rumour immediately. Rumours are treason.

Secret Societies

Some poor misguided citizens feel that The Computer does not provide them with enough guidance or leadership. They feel that membership of a highly illegal Secret Society is the only way to progress. These misguided citizens join secret societies for many different reasons. Some feel that The Computer should be doing things differently, or that The Computer should be overthrown, or that mutants should be stamped out (even registered ones), or that The Computer is doing a wonderful job and should be protected because Troubleshooters aren't effective. 

Membership of a secret society is - you guessed it - treasonous.

Communists

The Computer believes that Communists are its greatest threat. While it is tolerant of mutants (so long as they register themselves) and some secret societies (certainly those ones which believe that The Computer is doing a good job - although "tolerant" may be too strong a word for it), It is not tolerant of Communists. 

Should you discover you are a Communist, you are advised to terminate yourself. It would be a lot less painful than anything The Computer would have done to you.

Security Clearances

Everybody has a security clearance. Security clearance is based on colours of the spectrum. Citizens start out with Infrared clearance, which is low as you can get. These citizens wear black uniforms and are typically only allowed in Infrared areas unless specifically authorised by a citizen of a higher clearance. 

As time goes on, citizens who don't get terminated can be promoted to a higher clearance. The nine security clearances are Infrared, Red, Orange, Yellow, Green, Blue, Indigo, Violet and Ultraviolet. It is rumoured that there are security clearances above Ultraviolet. Rumours are, as you should know by now, treasonous. 

Not surprisingly, those citizens at the top are reluctant to let others rise to their level. Citizens at the top always have a close eye on those below them.  Citizens at the same level tend to compete to try to get promoted. Citizens at the bottom tend to try to eliminate those above them (and also the competition) in order to get promoted.

Availability of information is based on security clearance.  However, it is not unusual for vital information to be above your security clearance level.  For example, in the section on Mutations (see above), you are told that you should report any mutations you find.  However, information on mutations is above your current security clearance level.  This presents a dilemma - if you report a mutation, how do you explain knowing that it's a mutation?

The Computer takes security very seriously.  Citizens found possessing information or belongings above their clearance will be reprimanded.  Citizens in corridors above their clearance will also be reprimanded.  (Corridors are colour coded).  On extreme occasions, infrared citizens have been reprimanded for bleeding a colour above their security clearance.

Names

The Computer is a bit of a stickler for naming conventions. Every citizen's name contains their first name, their security clearance, the name of the sector they originated from (all sector names are 3 letters long) and their current clone number. 

Some examples are: 
 

Evap-R-ATE-1
Wond-R-FUL-2
Teela-O-MLY-1 (the famous vidscreen star) 
Mel-G-BSN-3
James-B-OND-1
What-U-WNT-5


etc, etc...

Infrared citizens leave the security clearance letter from their name, since technically they do not have any security clearance.  (And I stands for Indigo anyway).

Service Groups

Everybody in Alpha Complex belongs to a service group of some description.  These determine what job the clone did before becoming a Troubleshooter.  Some Troubleshooters are full-time, but still have strong links to the Service Group they worked for.  Other Troubleshooters are a bit like volunteer firemen - they work in the Service Group most of the time, and do Troubleshooting when The Computer needs them. 

Either way, a Troubleshooter's Service Group will determine what skills they started out with. 

The Service Groups are:

 

Internal Security	IntSec are the police of Alpha Complex.  While some officers openly display the fact they are in IntSec, many are undercover.  Troubleshooting teams may sometimes have IntSec officers in their midst, but they will always pretend to be from another service group.
Technical Services	Tech Services are the people who fix things around Alpha Complex.  They tend to have good knowledge of things electrical and mechanical, but they are unlikely to know how to reprogram The Computer (before you go getting ideas).  The roles of Tech Services and Power Services tend to overlap to some extent.
Research & Design	While Tech Services fix things around Alpha Complex, it is R&D who actually invent the things in the first place.  People from R&D tend to be reasonably good at fixing some things (but not as skilled as a Tech Services engineer), but tend to be better at doing research and coming up with novel ideas than their Tech Services counterparts.
CPU	Central Processing Unit look after the day-to-day running of The Computer, and tends to be rather bureaucratic.  Only the highest members of CPU are likely to have any computer programming skills though.  Having said that, CPU engineers tend to be skilled at using computer terminals to search for information quickly.
Power Services	Power Services look after the amenities (water, electricity, sewage, ventilation, etc) and associated cabling and piping within Alpha Complex.  Sometimes a Power Services engineer will know a route out of somewhere (or into somewhere) involving cabling ducts or ventilation shafts.
PLC	Production, Logistics and Commissary are responsible for agricultural and industrial production, along with the allocation of resources.  Troubleshooting teams normally visit PLC to be allocated equipment before going out on a mission.
Armed Forces	The army.  Their job is to provide Alpha Complex with military protection.
HPD&MC	Housing Preservation and Development & Mind Control look after the running of Alpha Complex when it comes to paperwork.  The HPD part involves making sure everybody has somewhere to live (and that they've filled out the right forms to get it).  The MC part ensures their minds are pure by providing quality vidscreen entertainment.  Think what it would be like if the bureaucrats ran all the TV stations.  That's what HPD&MC is like.

Bots

Alpha Complex is full of robots - more commonly called bots. Some Games Masters allow players to play bot characters instead of clone characters. Unlike clones, bots don't have security clearances as such. While this may seem like a good thing for players, bots tend to be treated worse than Infrareds. They also have some very strict rules regarding obeying orders.

Troubleshooters

Your character is a troubleshooter. This elite band of citizens (groups vary in size from 2 to 6, although groups are sometimes bigger) are supposed to track down trouble and shoot it. Sometimes they live up to their name and have trouble shooting. More often they find trouble in their own ranks and end up shooting each other. It is not unusual for an entire team to wipe itself out in a display of devoted loyalty to The Computer. (At least, thats what The Computer would like to believe it was). 

Invariably what happens is that some or all of the citizens belong to secret societies, who have given them instructions which quite often contradict the orders they were given by Troubleshooter HQ. Add some mutations to this mix (ones which would actually be useful, but the citizens daren't use them for fear of being terminated for being an unregistered mutant), and you start to see how silly a typical Paranoia mission can be.

A typical mission

The Troubleshooters normally get a mission alert first. This is a printed sheet of paper which tells them where to report for their briefing. Sometimes the mission alert will also give a few details of their mission. Occasionally these details may even be correct. 

Once the team has found their briefing room and briefing officer, they are given details of their mission. Missions normally involve a report of some trouble somewhere. The Troubleshooters normally have to go in, find the problem, find the cause, and fix it. This quite often involves shooting. 

Having been briefed, the Troubleshooters head off to PLC (Production, Logistics and Commissary) for some equipment. This will normally include laser barrels (since Troubleshooting teams tend to get through a lot of these) and various other stuff which The Computer thinks are appropriate for the mission. (Although the Troubleshooters will often find that PLC has lumbered them with stuff they simply want to get rid of). 

Next it's off to R&D (Research & Design) for some experimental equipment. The Computer believes that R&D is a valuable department and that they should be encouraged. While R&D do a lot of their work in the lab, there is no substitute for field testing. Troubleshooter teams are well suited for this sort of work since they are facing new challenges every day anyway. 

(It's rumoured that a high-clearance citizen in the Armed Forces managed to convince The Computer that Troubleshooter teams were better suited to R&D testing than members of the Armed Forces). 

Now it's time for the Troubleshooters to go off on their mission. Sometimes they never get this far, having managed to lose most of their clones on the way to R&D (or sometimes while in R&D). 

Eventually the citizens will return from their mission (or get wiped out). Should they return, they are debriefed. This quite often turns into a finger-pointing session where the Troubleshooters blame each other for various failures during the mission. Traitors are terminated. Loyal citizens are rewarded with credits, promotion, or both.