(covers information from several alternate timelines)
Inertial damping system, 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.
For USS Voyager to prepare to land on a planet, it was required for the ship to adjust its inertial dampers to maximum. Once landed, the ship released inertial dampers and adjusted them to match the planet's gravity. To depart from the surface, the inertial dampers were set to flight configuration. (VOY: "The 37's")
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 gastrointestinal 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")
In an alternate timeline, after the USS Enterprise was struck by the USS Bozeman in the starboard nacelle, the impact caused the Enterprise's inertial dampers to fail, and it to lose attitude control. (TNG: "Cause and Effect")
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")
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")
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.