Auxiliary power refers to the reserve energy on board starships which can be used in the event that the vessel's main power is disrupted or unavailable. The capacity of auxiliary power is limited and is intended for short term use only.
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, casuing several display screens on the bridge to display a blue alert message. (Star Trek V: The Final Frontier)
Auxiliary power can also be used as a means to strengthen certain systems at the discretion of the commanding officer. Shields, inertial dampers and structural integrity fields are 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 a possible implosion of a rift in 2369, he ordered Major Kira to route the auxiliary power into the shields and evacuate the pylons. (DS9: "If Wishes Were Horses")