In Star Trek starships, inertial dampers prevent crew members from being crushed by sudden acceleration.
A starship can accelerate, stop, turn, and enter warp without its occupants experiencing the full physical forces that such maneuvers would otherwise create. Inertial dampers are the fictional systems credited with reducing or cancelling those effects inside the vessel.
They are different from structural integrity fields. Structural fields protect the ship itself from stresses, while inertial dampers protect people and objects within the ship. Gravity plating also has a separate job: it supplies an artificial downward force for normal movement.
Episodes sometimes show inertial dampers failing or operating below capacity. Those failures provide a technical explanation for crew members being thrown across a bridge during combat. The concept helps reconcile dramatic starship maneuvers with the survival of the people aboard.