From Bulbapedia, the community-driven Pokémon encyclopedia.
Power points, or PP for short, are the energy that a Pokémon requires in order to perform a move.
Each move has a base number of power points. When a Pokémon learns a move, its current and maximum power points are set to this default value. Whenever the move is used, one PP is subtracted from the move's current power points. Once a move runs out of PP, it can no longer be selected as an attack. When the PP of all of a Pokémon's moves has run out, the Pokémon will be forced to Struggle. There is also a Pokémon ability called Pressure which, unless prevented by the ability mold breaker or the move Gastro Acid, will force the opponent to lose 2 power points instead of 1.
Under the power point system, a move may have a base number of PP equal to 1 or a multiple of 5 between 5 and 40. Up to three PP Ups can be applied to a move on a particular Pokémon, and each PP Up increases the move's maximum PP by 20% of its original value, unless the move originally had 1 PP, in which case it has no effect. A PP Max may also be used in place of three PP Ups. In Generation I and Generation II, the maximum PP value for moves that have a base PP value of 40 is 61 (likely due to a lack of data space); this was increased to the "proper" value of 64 in Generation III. Typically, the stronger or more desirable a move is compared to other moves, the lower its base PP. When the moves Shadow Rush and Struggle are used, PP is not deducted, effectively giving them infinite PP.
In Generation I, a glitch allowed struggling to be avoided by allowing the game to automatically use a move to attack, which could happen to any attack if a Pokémon attacked immediately after being defrosted, or due to a handful of moves (Bind, Clamp, Fire Spin, Hyper Beam, Metronome, Mimic, and Wrap) because of the auto-selection involved with partial trapping moves. A move that is used with 0 PP in this way will underflow to the maximum possible value (oddly 63, rather than the maximum PP Up value of 61), and full PP Up status will be applied to the move. In Generation II and onward, a check was added that prevents execution of a move that has 0 PP if it is autoselected.
Outside of battle, field moves like Fly, Dig, and all other HMs, may be used even if the current PP of the move is 0.
In Generation IV, the max PP of some moves has changed, such as Recover (from 20 PP to 10 PP) and Giga Drain (from 5 PP to 10 PP), among others. PP Ups applied during Generation III carry over; they use the new default value as a determinant for maximum PP, rather than the old one.
Elixirs, Ethers, and Leppa Berries restore a move's PP, and a visit to any Pokémon Center will replenish the PP of all of a Pokémon's moves to their maximum values.