There is one official rule: Survive; the last one standing wins.
There are a few more rules that play into the Hunger Games but they are relatively minor compared to the above:
Potential tributes are not to be trained in weapons or hand-to-hand fighting. This one is primarily to prevent the Districts' residents becoming proficient with weapons. It is also largely overlooked, as "Careers" (Districts 1 and 2, which are close to the Capitol and relatively well-cared-for overall) are groomed from childhood to compete in the Games, and there it's a great honor to be chosen (or to volunteer), instead of it being a death sentence like in many other Districts where the tributes grow up half-starved and beaten down before being chosen.
Once "launched" into the arena, you have one minute to gather your bearings before the Games actually begin. You may not leave your platform; you may not even move, until that time has elapsed. If you do, land mines will ensure you're not much of a factor in the rest of the Games.
While generally any method of obtaining food and other survival necessities is fine, including of course killing your fellow tributes to get their stuff, cannibalism is frowned on. While not technically against the rules, it's such a dishonorable insult-to-injury that the last notable tribute to do it was killed in a rather suspicious avalanche, probably set off by the Gamemakers to make sure he didn't end up Champion. In general, once a tribute is killed, other tributes in the area should back off to let the bodies be collected.
Unofficially, you have to remember that the Games are designed to be a show of the Capitol's power; the Districts are powerless to stop the Reapings, and are forced to watch the Games every year as punishment for long-past recriminations. In addition, survival generally involves winning donations from "sponsors", either wealthy Capitol residents or, sometimes, the entire population of a District. These gifts can be a life or death thing, and so every tribute strives to win the notice of those sponsors.
So, a few more rules generally shake out:
Be entertaining. These are the Games after all; they are held for the Capitol's enjoyment. If your time in the arena is boring, you won't get much camera time, you won't get much sponsor notice, and you won't get many gifts.
Be honorable. Dirty, underhanded tricks will keep you alive, but they won't win you sponsors. Nobody roots for the bad-guy wrestlers. This is more a guideline; the Careers don't follow it, and one of the longest-surviving "independents", a girl Katniss calls "Foxface", is one of the sneakiest competitors in the Games, and makes it to the Final Four.
Play the Games the way the Capitol has designed. Nothing you do in the arena can EVER look like you're mocking the Capitol. From time to time the Gamemakers will kill a tribute, just to remind everyone that they can, and the natural choices are the ones that run and hide, or play the game in a manner that might reflect badly on the Capitol (like the whole cannibalism thing).
The reason Seneca Crane, the Head Gamemaker of the 74th Hunger Games, was executed by the Capitol is because he didn't immediately blow Kat away as soon as it became evident what she was planning with the nightlock berries. She was trying to cheat the Capitol, either by getting both herself and Peeta out of that arena, or by ensuring the Capitol would have no Champion to dress up and make dance for the year following the Games.
Haymitch says that the Capitol hated him because he outsmarted them. He used the force-fields to kill people in a way not intended, and the Capitol killed his entire family after the Games.
During the 74th Hunger Games, there was a rule added to spice things up; if the last two tributes were from the same District, they would both jointly win the Games. This means that Katniss (and Haymitch) can get both herself and Peeta out of the arena alive, and they have to fight a truly allied pair of tributes from District Two (Clove and Cato), who have also survived to hear that particular announcement.
The rule was rescinded at the last minute, to give the audience the ultimate thrill; watching two tributes from the same District, two self-styled lovers even, fight each other to the death, or else watch one of them sacrifice themselves so the other can go home to be a pariah in their home District. Of course, Katniss turns this on its head by forcing a decision; either they both go home Champions, or nobody does.