To perform a band pushup, take a band and hold one end and take the other end in the other hand and wrap it across your back. Get into a normal pushup position and do pushups. The band overloads the top portion of the lift and really hits the triceps. Your body will be forced to explode off the floor to the top to beat the extra resistance at the top.
A mini band is a good place for most people to start with. A more advanced lifter could easily work up to a green band.
Band Pushups Chest Muscle Exercise - Start Position
Band Pushups Chest Muscle Exercise - Finish Position