In video games, first person means the player sees the game from the eyes of the character he or she is playing as. This is different than third person where the player watches what the character does from a in-game virtual camera. The player is supposed to feel like he or she is inside the character.
A common type of first person game is first-person shooter. In a first-person shooter the player can usually see his or her arms holding a gun or other weapon. Some games such as Call of Duty let the player look right down the sights of the gun to aim.
Many simulation games (or "sims") where the player controls a vehicle are first person. This includes most flight simulators and many racing video games. The player can often see the controls and instruments of the vehicle, such as the steering wheel and speedometer in a car or the joystick and altimeter in an airplane.