Inertial dampers were a system used on almost all starships to counter the effects of rapid acceleration and deceleration of a starship 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") It also could not jump to warp with external inertial dampers disengaged. (Star Trek)
In 2155, while using a comet as camouflage to infiltrate John Frederick Paxton's base on Mars, Travis Mayweather disabled the inertial dampers of the shuttlepod, causing it to maintain an erratic flight path to better resemble a chunk of the comet. This caused Malcolm Reed some distress, as he had a history of gastro-intestinal problems. (ENT: "Terra Prime")
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 (while saving a Changeling that later died) was killed when their inertial damping system apparently failed as the ship went into warp. (DS9: "The Ship")
On stardate 50482.2, a cascade virus previously planted in the computers of the USS Defiant by Maquis defector Michael Eddington wiped all systems aboard. Several days later, the inertial dampers had not been fully repaired and, while the ship could reach warp 6, Chief O'Brien added, "Don't expect a smooth ride". (DS9: "For the Uniform")