The red colored flag is a special indicator used by coaches to inform referees on the field that they would like the previous play reviewed. A coach in the National Football League is allowed to challenge specific calls made by referees during the course of the game. Each coach is given two challenges per game, and must have a timeout remaining in order to use it. If the coach uses and is successful on two challenges, they are given a third. If a coach wishes to challenge a play, they must throw their red flag onto the field prior to the next play occurring.
The reason the NFL decided on a red flag was because it was the most distinguishable from the yellow flags the referees carry. The system was implemented to allow only the head coach of a team to challenge a play, therefore they are the only one with a red flag. During the final two minutes of each half, neither coach is allowed to throw their flag and challenge a call.