In 2265, after their warp engines were burned out due to an encounter with the galactic barrier, the crew of the USS Enterprise adapted the power packs used at the lithium cracking station on Delta Vega to regenerate them. (TOS: "Where No Man Has Gone Before")
Phaser power packs, used in some type 2 phasers and phaser rifles, were one of the most common types. In the 23rd century, members of a landing party when carrying type 2 phasers would also carry one or more "reserve" power packs. These packs were designed to be quickly "swapped out" to "reload" the phaser.
Another type was typically used as an emergency power source on-board starships for use when main power was off-line. They could be used, among other things, to "unlock" closed doors. (TNG: "Starship Mine")
A third "power pack" was a component of the aqua-shuttle that was required for operation of the vessel's phasers. In 2269, the power pack was damaged when the aquashuttle carrying members of the USS Enterprise crew was attacked by a sur-snake during a mission on the planet Argo. (TAS: "The Ambergris Element")