AIM-9 Sidewinder and AIM-120 Scorpion are mounted on LAU-7 or LAU-127/8 launchers, known as "rails".
Upon signal from the pilot, the motor fires. The missile goes forward.
The umbilical cable connecting the missile to the launcher breaks away as the missile moves forward.
As far as how they're held on, there is a spring loaded lock mechanism, that is overcome by the missile motor.
pg 3-22 here:
DETENT, DETENT LOCK, DETENT-LOCKSOLENOID, AND SNUBBERS.—The detent, detentlock, detent-lock solenoid, and forward snubbers (fig.3-18) make up an electromechanical system that restricts longitudinal and lateral motion of the mounted missile. This system prevents accidental launch or release during catapult takeoff or arrested landings. A slotted detent wrench/safety pin is installed through the mechanism assembly to prevent movement of the aft detent and to safe the launcher firing circuits. It is also used to raise the aft detent lug sufficiently to clear the missile hanger during loading. The slotted detent wrench/safety pin is removed before flight. When the detent is down (normal), the striker points make contact with the buttons (contacts) on the forward hanger of the missile. The forward striker point does not have a function for the missile; the aft striker point makes contact between the missile and the firing safety switch
Body mounted missiles such as an AIM 7 or AIM -120 on a F-15 are kicked out away from the fuselage a couple of feet before the missile motor activates. There is a long cable that breaks away once the missile starts to leave.
The ejector foot is actuated by a small explosive charge.