After owning a Pokémon for a certain amount of time, it may change into a different Pokémon. This is called evolution.
Most Pokémon evolve once in their lifetimes, some evolve twice. Some don't evolve at all.
A Pokémon can evolve one of three ways, the simplest way is leveling up. If your Pokémon wins a battle, it gains experience points. After it has enough experience points, it will "level up", or grow to a higher level. This means it has more power and will be stronger.

After a Pokémon has reached a certain level, it will evolve. There's no specific way to tell at what level a Pokémon will evolve, but usually, similar types evolve at the same level. For example, a Weedle will evolve into a Kakuna at the same level a Caterpie evolves into a Metapod (7), and a female Nidoran will evolve into Nidorina at the same level a male Nidoran will evolve into Nidorino.

Another way for a Pokémon to evolve is to use a special Stone on it. Stones can be found lying on the ground, and can also be bought on the 4th floor of Celadon Department Store.
Using a Stone on a Pokémon is really easy. Just get a Stone that matches your Pokémon's type (Leaf Stone works on Grass Pokémon, Thunder Stone works on Electric Pokémon, etc.), then just select it in the Item menu, and select the Pokémon you want to use it on. Remember, not all Pokémon are affected by the Stones. Some will evolve on their own when their experience reaches a high enough level.

There's one type of Stone you can't buy at Celadon Department Store. It's the Moon Stone. This Stone works on Pokémon that came from the moon, like Clefairy.
How to tell if your Pokémon comes from the moon? My advice: Look at it. If it doesn't even vaguely resemble a real life animal, it probably came from the moon.
For example, Ponyta looks a lot like a horse, so it probably didn't come from the moon. Psyduck looks a lot like a duck, so it probably didn't come from the moon. But now, take a look at Jigglypuff. It's round, probably a little fuzzy, and look at those eerie green eyes! A sure space candidate. Try using a Moon Stone and see if it has any effect.

The third way for a Pokémon to evolve is by trading it. If you trade Kadabra, for example, it will evolve into Alakazam. Only 4 Pokémon evolve this way. They are: Kadabra-Alakazam, Graveller-Golem, Haunter-Gengar, and Machoke-Machamp.

If your Pokémon evolves, it will become more powerful, and it's elemental attacks will be more effective. Elemental attacks include Fire, Water, Air, etc.

If your Pokémon starts evolving and you don't want it to, press the B button and it will stop. After it gains another level, it will attempt to evolve again. The advantage to this is that the "unevolved" Pokémon will learn attacks sooner. For example:

Zubat evolves at level 22. If you stop it from evolving, it will learn its next attack (Wing Attack) at level 28, and its final attack (Haze) at level 36. But if you let it evolve, it will learn Wing Attack at level 32, and Haze at level 43!