Force fields are invisible electric fields set up by the Capitol that are used to prevent tributes from escaping the arena and any other places so they won't run away from the Games. It tosses anything that hits it back to where it was before, so tributes couldn't commit suicide by jumping off the roof of the Training Center. It was also used to protect Gamemakers from tribute attacks.
50th Hunger Games
Haymitch Abernathy won the The 50th Hunger Games by using the forcefield to defeat the female from District 1 by letting her throw her axe into it and watching it come back and lodge itself in her head. He was at "His cliff", a sharp drop off on a rocky ledge that he discovered shortly before Maysilee's death. This stunt crowned Haymitch victor, but President Snow didn't like his stunt, so he killed Haymitch's family and girlfriend.
74th Hunger Games
The first time that Katniss Everdeen ever comes in contact with a force field is on the Traning Center's roof with Peeta, who shows her how the force field works by tossing an apple into it and catching it when it is thrown back.
75th Hunger Games
In the Training Center while Katniss was preparing for the 75th Hunger Games, Beetee and Wiress pointed out the weak spot in the force field that is in between the Gamemakers and the tributes to her, which is a small wavering square in the air, revealing the invisible force field. Katniss thinks that this field is in place because of her private session the year before.
It is also mentioned that on the day before their interview, Katniss and Peeta spend their day on the Training Center's roof playing a game where Katniss would throw the ball into the force field and Peeta would catch it when it came back.
In the 75th Hunger Games, Peeta almost died from getting shocked by a force field, but luckily Finnick revived him with mouth to mouth. Blight, the male tribute from District 7 also hits this force field, but Finnick isn't present to revive him so he dies. Also, Katniss got stuck behind a force field for an hour when the Jabberjays attacked. Katniss used the wire Beetee planned for a trap and a lightning bolt a Gamemaker created to destroy the forcefield around the arena.
This was done by firing an arrow with Beetee's wire attached to it with Katniss' bow into the weak spot - the chink in the armour of a forcefield.