Vehicle Missions are those where you get into a specific vehicle and press a certain button to being a mission unique to that vehicle. For example, the Ambulance has a Paramedic mission tied to it that can be started by pressing a specific button once inside. These missions usually have a certain quota to reach and when it is, a reward is given to Carl Johnson.
To start a vehicle mission, press the right analog stick in on the PlayStation 2 or Xbox, or the 2 key on the PC version.