Inertial dampers are a system used on almost all starships to counter the effects of rapid acceleration/deceleration of a starship by sustaining and absorbing the natural inertia of a vessel as it either moves through space or if it is under attack from another vessel.
A starship cannot 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 are activated by default, the crew can enable/disable and configure the strength of them manually. If the inertial dampers are disabled, they can 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 can exaggerate the damage sustained by a starship in a battle as the ship rocks more violently when its inertial dampers are disabled. (TNG: "The Chase", "Gambit, Part II")
The entire crew of a Jem'Hadar attack ship (while saving a Changeling that would later die) was killed apparently when their inertial damping system 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 dampeners 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")
Our signature drink, a frozen ginger lemonade with or without vodka concoction of lemonade, ginger ale, and optional vodka, on the rocks and garnished with crystallized ginger, has been dubbed the Inertial Dampener (by me).