The Hunger Games
The Hunger Games are a competition in which each of the twelve districts of Panem are required to send two tributes - a male and a female - from the ages of twelve to eighteen off to an arena to participate in a televised fight to the death. There is only one known exception to this, which is the Quarter Quell in Catching Fire. In these particular Games there were tributes of all ages, since they were taken from past victors. See the 'Quarter Quell' section below for more details. The Games are mandatory watching for the people of the districts, and people of the Capitol, who are safe from participation, often enjoy watching these cruel games.
 The Reapings
Tributes are the children who are sent to participate in the Hunger Games. The process of choosing tributes is called a reaping, and constitutes two bowls - one full of slips with boys' names, and one with girls' names. One slip is pulled from each bowl, and the person whose name is called is the tribute for that District. Effie Trinket, the escort for District 12, is described in the book as always calling out the girls' name first, but it is unconfirmed whether this is true for all the districts.
Once a tribute has been announced another eligible person of the same gender may volunteer in their stead. This is very unusual for districts such as District 12, so when Katniss Everdeen volunteered for her sister Prim people were rather taken aback. In wealthier districts such as District 2, however, the people tend to raise their children as contenders for the games, and there is often complications regarding volunteering since so many are willing.
Every person - whether or not they are eligible to compete - is required to go to the square on reaping day. Only people on "death's door", as described in The Hunger Games are exempted, and officials go around the Districts during the reapins to enforce this. Anyone found in their houses who is not about to face death is imprisoned. The reapings are also televised - as is most of the Games - and all of the Districts are required to watch. The reapings in every district occur at different times, so theoretically a person could watch them all in a row. However, since most people have to attend reapings themselves, this is really only possible for people of the Capitol. The reapings are re-played later that night so nobody misses any.
The number of entries each possible tribute has in their respective balls depends on their age. For example, someone who is 12 (such as Prim) would only have their name in once, whereas someone who is 18 (such as Gale) would have their name in 7 times. Therefore, there are greater odds of being picked the older you are. This did not seem to help Prim however, since her single piece of paper was drawn from the ball out of thousands.
It is important to note however, that it is possible to have your name in more times, if you've taken out tesserae. A tessera is a supply of grain and oil that will suffice for one person for a year, though it is still not very generous. However, it is better than nothing and any child between the age of 12 to 18 can take out one tessera for each member of their family. However, each tessera they take out results in one more entry in the ball. As well, they cumulate each year. For example, Katniss took out three tesserae as soon as she turned twelve (for her mother, Prim and herself), so had four ballots in the ball. The next year she took three out as well, so ended up with eight ballots.
The tesserae really only affect the really poor families, such as the families of Gale and Katniss, or other people from the Seam in District 12, or similar areas in the other districts. This is how Gale has 42 ballots when he is 18, the year Katniss first enters the Games. They are just another method of targeting the poor families, likely since they are not as valuable as the merchants' kids, who can themselves become merchants and support the Capitol. Once again, it is all about greed and control.
The Hunger Games came into existence as a result of the Dark Days, a time when there was an uprising by the districts against the Capitol. Distrcits 1 through 12 were defeated, while District 13 was believed to be completely obliterated by nuclear warfare. To remind the people of Panem that a rebellion must never occur again, the Capitol holds the Hunger Games each year. It serves to show just how much control the Capitol has over the districts, and that they districts couldn't fight back even if they wanted to.
The districts, in addition to being forced to watch the Games as outline above, are also required to treat it as a festivity. The Capitol makes it out to be an honour to be chosen tribute (and in Districts like 2 where they are consistently volunteering it appears to be a great honour for sure), and grants a life of luxury to whoever is able to win it. This includes a spot in the district's Victor's Village, and special favour shown to the winning district for the entire year.
There are a number of things that happen each year before, during and after the Hunger Games. First of all of course are the reapings (as explained above). There also is the first official introduction of the tributes to the public, interviews, public training and private training that occur before the Games themselves.
Traditionally, the newly named tributes are allowed one hour for good-byes before leaving their districts for the Capitol. In The Hunger Games this is when Katniss receives the mockingjay pin from Madge Undersee and the bag of cookies from Peeta's father, among other good-byes.
However, for the 3rd Quarter Quell that happens in Catching Firethe rules are changed and Peeta and Katniss are not allowed any time before behing shipped off to the Capitol. Katniss never has the chance to say good-bye to her mother, Prim, or anyone else in the district. She had specificly planned some things to say to Gale Hawthorne, about how much he meant to her over all the years, et cetera, but she never does. Considering Katniss plans on not returning (since she wants Peeta to win), this is a very hard blow.
 Quarter Quells
Quarter Quells are a very special kind of Hunger Games. They occur once every 25 years, and are designed to be even worse than the other years, as an even more harsh reminder of the control the Capitol has over the distrcts. In Catching Fire Quarter Quells are described as "a glorified version of the Games to make fresh the memory of those killed by the districts' rebellion." Weeks before the reaping a card is read on television (mandatory watching, of course), which explains the twist for that year's Quarter Quell.
The first Quarter Quell was held 25 years after the Dark Days. That year's card read "On the twenty-fifth anniversary, as a reminder to the rebels that their children were dying because of their choice to initiate violence, every district was made to hold an election and vote on the tributes who would represent it." Katniss comments that this would be much worse, since you would know it was your own community who choose for you to go rather than a product of a random (if biased towards the poor) draw.
The second Quarter Quell was the 50th Hunger Games, and it was the year Haymitch Abernathy won. This Games forced the districts to give up twice as many tributes as usual - that is two males and two females. The Card for those Games stated that this was a reminder "that two rebels died for each Capitol citizen." These Games would also have been a horrible one, because you're chances that year would be half of that in other years, and in the end more than twice as many children die. Mrs. Everdeen's friend Maysilee Donner was one of the female tributes that year.
The third Quarter Quell; the 75th Hunger Games, is featured in Catching Fire. It requires the tributes to be taken from the pool of existing victors, to remind the "rebels that even the strongest among them cannot overcome the power of the Capitol". To Katniss this is devastating. For her, one of the only good things she took from the Games was that by having participating once ensured she never had to again. Victors are ommitted from the reapings, so Katniss never expected she would have to go back.
There were only ever four victors from District 12, and the only alive being Katniss, Peeta and Haymitch. This meant Katniss, as the only female victor, was definitely going back, and that either Peeta or Haymitch were. Even if Haymitch's name was drawn, Peeta planned on volunteering to take his place. The thought of she and Peeta both returning to the arena made her sick, and Katniss actually got drunk (with Haymitch) for the first time ever.
Although the slips containing the details for each Quarter Quell were said to have been written out after the Dark Days, Katniss has suspicions that the third, at least, was specially designed by President Snow, because of how Katniss had failed on the Victory Tour to convince him that she was desperately in love with Peeta, and therefore dull down the rebellion. When that plan failed, he threw her back in the Games. Not only would Katniss likely die in another Games, since the victors were indeed the strongest (having survived the horrors of the arena and been granted practically lives of luxury in the Victors Village of their respectful districts) killing twenty-three of them would help kill the small rebellions that were happening all over Panem. Though its never proven that the card was made by Snow himself rather than 75 years pervious, this seems a very likely possibility. It was just too perfect for the current situation, even in Katniss' mind.