Can someone explain to me what kind of government The Hunger Games contains?
This is a excellent, but difficult question to answer. Despite the name President for the leaders of the government, the nation of Panem is most definitely not a democracy. If you read on to Mockingjay, the government is discussed and when a possible new government based on the Republic of Rome is proposed by Gale, Katniss immediately dismisses it for its foreigness.
The trick to this question is that the citizens of the Capitol have a different set of rules than the rest of the districts. The basic thought that most readers would have involving the government would be that it is a totalitarian communist government. The government controls the means of production in all the districts, and the people lack the freedom to enact change in the government and have few civil liberties. The problem with that answer is that the people of the Capitol seem to abide by a different set of rules.
Therefore, the best answer is to look at Panem as an empire focused on the Capitol with the districts being colonies in a model of mercantilism. The citizens of the Capitol enjoy a great life based on the subjugation of the peoples of colonies under the control of the mother country. You could liken President Snow to an English Monarch and the districts as colonies like the 13 American Colonies had the American Revolution been unsuccessful. While a "Hunger Games" would have been unlikely, it is certain that severe punishments would have been enacted while still expecting the allegiance of the colonists.
Dictatorship in a distopian society
I would describe the government in the Hunger Games to be a form of Dictatorship.
Dictatorship is when one person or a small group of people has all the power of the country in their hands. They can control the country in anyway they like. It is not like Monarchy, though one person is ruling, there are not laws are put in place to protect the freedom and rights of people. Dictatorship can take much of the freedom away from it's subjects. In the book "the Hunger Games" you see that president Snow has all the power over all the districts. When Snow wants someone killed it happens, when he wants the Games they happen, what he says goes. That's how dictatorship works. It's also a lot like the situation in North Korea..