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There's one main rule that is known: Survive.

There are a few unstated rules that Katniss is aware of and relays to the reader:

• Be entertaining


• Make Capital look good/magnanimous/etc.


• Demonstrate total subservience and obedience to Capital

Formally, though, once in the arena.. Well, to quote Katniss:

There are no rules in the arena, but cannibalism doesn't play well with the Capitol audience, so they tried to head it off.

There are some related rules, but not much is ever said about them. Nor, in fact, are they really rewarded the way you are led to believe they will be. (See book 3 and details about any of the previous winners to understand.)

An example of a related rule, though, (referenced, rather than directly stated) is:

The tributes from 1, 2, and 4 traditionally have this look about them. It’s technically against the rules to train tributes before they reach the Capitol but it happens every year. In District 12, we call them the Career Tributes, or just the Careers. And like as not, the winner will be one of them.

Lots of people are forgetting one, which is that you are allowed one token to bring into the arena, which cannot be used as a weapon. In the book it describes people have tried to sneak poison in as a weapon through their token.

The rules are simple.

1. Last one standing wins.


2. Don't train for it, although that one doesn't seem to be enforced.


3. Don't make the capital look stupid.

The last one covers a lot of ground, and isn't a formal rule, but if you break it, you could pay for it for the rest of your life.

There are many rules.

1. You cannot step off your pedestal at the beginning of the Games before the minute is up (if you do you'll blow up);


2. Tributes are not allowed fighting in the training area; if they do, they will be punished;


3. Potential tributes are not allowed to train before being Reaped - this one isn't very enforced, though, thus the Careers;


4. Do not isolate yourself at the edge of the arena - as seen in the first novel, when Katniss is almost at the edge of the arena, kilometers away from the rest of the tributes, the Gamemakers start a forest fire to turn her back. I believe that this rule was put in place after Haymitch won his Games by using the force field around the edge of the arena to kill the remaining tribute;


5. As stated by Haymitch but technically not an official rule: Don't die.

There's one other rule. The rules can be changed in the middle of the game. Capricious rule changes are another way the state demonstrates its absolute power.

1. A boy and a girl between the ages of 12-18 will be presented in a public "reaping," where they will represent their districts as tributes.




2. Tributes must wait a full 60 seconds before stepping off your plate. If you step off before then, you're legs will be blown off. Once the 60 seconds are up, the game begins.




3. Don't offend the Capitol. This is one of the many implied rules, as it is not stated but obvious. If the Capitol feels that a Tribute has made a mockery of them in any way, they will suffer.




4. Tributes are allowed one token to bring with them to the games. However, the tokens will be checked to ensure that they cannot be used as a weapon, or contain any other dangerous elements such as poison.




5. There will be no canniballizm in the games. This rule is implied, as in one of the earlier games a tribute ate its victims, and the gamemakers killed him.




6. Tributes are not permitted to fight while in training. If they do do, there will be consequences. All grudges held against another tribute will have to be saved until in the arena.




7. The tributes (with the exception of the Careers) are not aloud to train before being reaped for the Hunger Games. Consequences are severe.



8. 
   

   Survival. This is a fight to the death. A lone Victor must remain.
   Note: the rule has been an exception for one game; the 74th Hunger Games.

   
   
   

             Good luck, and may the Odds be Ever in your Favor...
   
   

   
   

Below I have collected everything relating to the Games described with the term rule throughout the three books. Obviously, some of them have already been mentioned in other answers, but some have not (including the one passage that is the only clear and deliberate explanation of the rules by Katniss). Some are more directly related to the Games themselves, while others are more about the peripheral activities that go along with the Games. Some might not precisely be official rules, but they are described as rules nonetheless. Similarly, there might be other rules that are not described as "rules". I have presented them in the order they appear in the books.

• 
  

  And even though the rules were set up by the Capitol, not the districts, certainly
  not Madge’s family, it’s hard not to resent those who don’t
  have to sign up for tesserae.

  
  




• 
  

  The rules of the Hunger Games are simple. In punishment for the uprising, each of the twelve districts must provide one girl and one boy, called tributes, to participate. The twenty-four tributes will be imprisoned in a vast outdoor arena that could hold anything from a burning desert to a frozen wasteland. Over a period of several weeks, the competitors must fight to the death. The last tribute standing wins.

  
  




• 
  

  The rule is that once a tribute’s name has been pulled from the ball, another eligible boy, if a boy’s name has been read, or girl, if a girl’s name has been read, can step forward to take his or her place.

  
  




• 
  

  “No,” says Haymitch, stopping him. “Let the bruise show. The audience will think you’ve mixed it up with another tribute before you’ve even made it to the arena.”
  
  “That’s against the rules,” says Peeta.
  “Only if they catch you. That bruise will say you fought, you weren’t caught, even better,” says Haymitch.

  
  




• 
  

  It’s technically against the rules to train tributes before they reach the Capitol but it happens every year.

  
  




• 
  

  “Wouldn’t you love to pull her back out here and get a response?” Caesar asks the audience. The crowd screams assent. “Sadly, rules are rules, and Katniss Everdeen’s time has been spent. Well, best of luck to you, Peeta Mellark, and I think I speak for all of Panem when I say our hearts go with yours.”

  
  




• 
  

  There was a guy like that a few years ago from District 6 called Titus. He went
  completely savage and the Gamemakers had to have him stunned with electric guns to collect the bodies of the players he'd killed before he ate them. There are no rules in the arena, but cannibalism doesn't play well with the Capitol audience, so they tried to head it off. There was some speculation that the avalanche that finally took Titus out was specifically engineered to ensure the victor was not a lunatic.

  
  




• 
  

  There’s been a rule change in the Games. A rule change! That in itself is mind bending since we don’t really have any rules to speak of except don’t step off your circle for sixty seconds and the unspoken rule about not eating one another. Under the new rule, both tributes from the same district will be declared winners if they are the last two alive.

  
  




• 
  

  I nod because I do understand. About owing. About hating it. I understand that if Thresh wins, he’ll have to go back and face a district that has already broken all the rules to thank me, and he is breaking the rules to thank me, too. And I understand that, for the moment, Thresh is not going to smash in my skull.

  
  




• 
  

  “Greetings to the final contestants of the Seventy-fourth Hunger Games. The earlier revision has been revoked. Closer examination of the rule book has disclosed that only one winner may be allowed,” he says. “Good luck and may the odds be ever in your favor.”

  
  




• 
  

  The beauty of this idea is that my decision to keep Peeta alive at the expense of my own life is itself an act of defiance. A refusal to play the Hunger Games by the Capitol's rules.

  
  




• 
  

  I'm sorry, Peeta, I think. I'm sorry I couldn't save you. Save him? More likely I stole his last chance at life, condemned him, by destroying the force field. Maybe, if we had all played by the rules, they might have let him live.

  
  

There are no rules for the Games. But certain choices have certain consequences.

For example, Katniss Everdeen and Peeta Mellark with the poison berries. The consequence of that choice was District Twelve being obliterated just as District Thirteen was. They humiliated the Capital. That was their consequence.

Generally, the Quarter Quell has rules that give the Games a sadistic twist, like voting for the tributes, double the amount of tributes, and returning victors, which is not supposed to happen in regular Games.
There is one unspoken rule in the regular Hunger Games, which is no cannibalism, because Titus apparently freaked people out by eating his victims' hearts.
And there can only be one victor, as that was in the "rule book." They said there was "further inspection," remember? So obviously the amount of victors is a rule (that Katniss and Peeta shattered).

There are probably some rules written by the capitol for the Hunger Games, but we never get to see exactly what they are. You can know this because in Katniss’ Hunger Games, Claudius Templesmith, the announcer says
There has been a rule change. This implies that there had to be rules in the first place such as there can only be one winner. Other rules include that tributes must be at least twelve but no older than eighteen and they are permitted to bring one token from their district as long as that token give them no unfair advantages in the games. Despite these rather obvious rules, there are as Katniss says, “unwritten rules.” However the unwritten rules mostly just translate to do not be a complete animal and do not cause trouble with the capital. Actual examples of these unwritten rules are “don’t eat other tributes dead bodies,” and “let the capital collect the dead.”