Inertial dampers, or inertial dampeners, were a system used on almost all starships to counter the effects of rapid acceleration and deceleration by sustaining and absorbing the natural inertia of a vessel as it moved through space or if it was under attack from another vessel.
A starship could not jump to warp speed without inertial dampers, as the rapid acceleration would smash the crew into the walls, killing them instantly. (VOY: "Tattoo")
Although the inertial dampers were activated by default, the crew could enable, disable, and configure the strength of them manually. If the inertial dampers were disabled, they could be reactivated by resetting the inertial dampening field base at three-zero-zero and re-initializing the dampers. (VOY: "The Cloud")
This use of them could exaggerate the damage sustained by a starship in a battle, as the ship rocked more violently when its inertial dampers were disabled. (TNG: "The Chase", "Gambit, Part II")
As a safety precaution, the inertial dampers on Federation vessels had an automatic bypass that kicked in when the system was disabled. (TNG: "Cost of Living")
The entire crew of a Jem'Hadar attack ship was killed when their inertial damping system apparently failed as the ship went into warp. A Changeling that was aboard initially survived, but later died of its injuries. (DS9: "The Ship")
According to the Star Trek Encyclopedia (2nd ed., p. 205), inertial dampers were "invented" by Star Trek writers to explain how the crew avoided becoming "chunky salsa" when starships accelerated or decelerated.
The external inertial dampener from the film Star Trek was described as a "field-manipulation device designed to compensate for gravitational or acceleration forces when a starship is in dock" by the Star Trek Encyclopedia (4th ed., vol. 1, p. 256).