A booster was a device that augmented the capability of another device.
In 2267, the auditory sensor of the USS Enterprise was augmented with a booster which increased its abilities by one to the fourth power. This enabled it to hear every sound on board the ship, including the heartbeats of a skeleton crew, the court martial panel, and a presumed-dead Benjamin Finney. (TOS: "Court Martial")
When the Arcos was found on the surface of Turkana IV, Geordi La Forge suggested to use the ship's sensing device and install a booster. That would relay the signals to the USS Enterprise-D and allow the crew to track the life signs on the planet. (TNG: "Legacy")