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 was also used to protect Gamemakers from tribute attacks.
The first time that Katniss ever comes in contact with a force field is on the Training Center's roof with Peeta, who shows her how the force field works by putting his hand onto it and feeling the sharp zap that follows.
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 Katniss' 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. Katniss uses the wire Beetee planned for a trap and a Gamemaker-created lightning bolt to destroy the forcefield arena. This was done by firing a knife with Beetee's wire attached to it with her bow into the weak spot - the chink in the armour of a forcefield.