An induction coil was a coiled loop of wire. Running an electrical current through such a coil created a magnetic field that opposed any changes in the current flowing through the wire.
Zefram Cochrane's original warp core included an induction coil. (Star Trek: First Contact)
In 2373, Neelix repaired a Nezu orbital tether carriage by replacing its damaged induction coils with attitude control thrusters from one of USS Voyager's shuttles. (VOY: "Rise")
When Liam Bilby acquired three Klingon disruptors for the Orion Syndicate in 2374, their induction coils were burned out. Via Starfleet Intelligence, Miles O'Brien got replacement coils from the Klingon ambassador on Farius Prime and repaired the disruptors. (DS9: "Honor Among Thieves")
Klingon cloaking devices, such as the IKS Rotarran's, used induction coils. In 2375, while he and Rom were stealing the device, Quark touched the coils, giving himself a mild electric shock. Rom warned him about them after that happened. (DS9: "The Emperor's New Cloak")
Prior to the Battle of Cardassia, Chief O'Brien told main engineering to re-align the induction coils to improve the maneuverability of the USS Defiant. (DS9: "What You Leave Behind")