Auxiliary power, or aux power, referred to the reserve energy on which starships, satellites, space stations, and planets could use in the event that the main power was disrupted or unavailable. The capacity of auxiliary power was limited and was intended for short term use only. On the largest scale, on planets, auxiliary power was generated in reserve banks.
In the mid-2270s, after a plasma energy bolt fired from V'ger overloaded the engineering systems of the USS Enterprise, resulting in a seventy percent drop in deflector power, Captain James T. Kirk ordered that auxiliary power be diverted to the deflectors. (Star Trek: The Motion Picture)
In 2285, when the USS Enterprise was attacked by the USS Reliant under the control of Khan Noonien Singh, heavy damage was done to the engineering section such that main power was totally unavailable. Auxiliary power was brought on line as soon as possible by chief engineer Montgomery Scott, but, as Captain Spock said "One thing is certain, we cannot escape on auxiliary power." (Star Trek II: The Wrath of Khan)
In 2287, following an attack by a Klingon Bird-of-Prey commanded by Klaa, the USS Enterprise-A was reduced to auxiliary power, causing several display screens on the bridge to display a blue alert message. (Star Trek V: The Final Frontier)
In 2293, during the Battle of Khitomer, a Bird-of-Prey commanded by General Chang fired a photon torpedo at the Enterprise-A's secondary hull, causing sparks to erupt across the bridge. When Captain James T. Kirk ordered switching to auxiliary power, leading Spock to tell him "Auxiliary circuits destroyed, Captain." while presumably trying to fix them. (Star Trek VI: The Undiscovered Country)
Auxiliary power could also be used as a means to strengthen certain systems at the discretion of the commanding officer. Shields, inertial dampers, and structural integrity fields were often supplemented with auxiliary power.
The same year Commander Benjamin Sisko asked Chief Miles O'Brien if there was enough time to transfer the auxiliary power into the deflectors while the space station was pulled into the wormhole but O'Brien answered that there was not enough power to transfer. (DS9: "Q-Less")
When Commander Sisko prepared the station for the possible implosion of a subspace rupture in 2369, he ordered Major Kira to route the auxiliary power into the shields and evacuate the pylons. (DS9: "If Wishes Were Horses")
Seconds before the Enterprise-D's primary hull crash landed on the surface of Veridian III in 2371, Data announced that he had rerouted auxiliary power to the saucer's lateral thrusters in an effort to level the starship's descent. (Star Trek Generations)