Level
Levels are used in all of the Pokémon RPGs as a measurement of how strong a Pokémon currently is. It is determined by how much experience that Pokémon has. A Pokémon's level will range from 1 to 100. When a Pokémon gains a level, its stats increase by a small amount. It may also learn a new move or evolve.
A Pokémon's level will also affect its happiness, raising it slightly when the Pokémon levels up. This means that a Pichu trained in battle will eventually evolve into a Pikachu, with its happiness going up slightly each level. However, this is not the case with using Rare Candies, and a Pokémon that evolves by tameness will usually not evolve through their use, unless, of course, its tameness rating is significantly high already.
Also, when a Pokémon has reached level 100 it cannot continue gaining Effort Points in Generations III and IV. In Generations I and II, however, the player can use the box trick.
By doing the Missingno. trick or using other cheats, a Pokémon can temporarily acquire a level higher than 100. However, whenever this Pokémon gains any amount of experience, its level will revert to 100.
In the Anime
Levels are famously ignored completely in the anime. Various references have been to "power levels" but this seems to be more of a general term, rather than the levels used in the games. A classic example of this is Ash's Pikachu. After appearing in more than 500 episodes in a row, Pikachu would logically be level 100, however, Pikachu is still often beaten by other Pokémon. One effect of this is that Pokémon in the anime learn moves that would normally be hard to get. For example, in its debut episode, Dawn's Piplup used Bide, a move Piplup normally doesn't learn until level 18, when all starter Pokémon are obtained at level 5. Therefore, level up moves can generally be learned at any time in the anime.
However, in the two Mystery Dungeon specials, levels are mentioned, but not explained. This may mean that Pokémon themselves understand the concept of levels in the anime but humans do not.
Trivia
Before Generation IV, it was impossible to get a Pokémon at Level 1 without the aid of a glitch or cheating device.
