First you should have some background information: the Hunger Games were established at the end of the Dark Days when District Thirteen held an uprising and rebellion against the Capitol. The Games are a punishment for the rebellion and a warning for future residents of Panem that something worse may come their way if they rebel again.
The Reaping happens once a year, a few months before the Games begin. A person's name will automatically be submitted into the Reaping once he or she is twelve years old, and will no longer be submitted under one of two conditions: 1) Said person has turned 18, and is no longer eligible for the games, or 2) they had been Reaped already and won the games. People will also have a chance to enter their names into the Reaping more than once in exchange for goods.
A representative from the Capitol - in this case, Effie Trinket - to Reap the tributes. A short video about the history of the games will play, and then Effie will then draw the names of the two unlucky people. First she will pull the girl's name, and the tribute will walk up to the stage to stand next to Effie. Then, she will draw the boy's name, and he will do the same.
Others may also volunteer as tribute in the games as a replacement of the person who had been Reaped. Katniss had volunteered for her little sister, Primrose (Prim), because she knew that Prim had no chance of surviving in the arena. In District Twelve, volunteers are rare, but in the more luxurious districts, such as District One or Two, volunteers are much more common, because they had been training for the Games, also known as Careers.
The two tributes will have a short time with people they are close to before they head for the Capitol, and then they go to a training center where they spend the next few weeks getting ready for the games.