Evolution
Pokémon have always evolved, right from the start. They usually evolve into some creature that is some way related, perhaps in color, or in being. There are many ways of evolving your Pokémon. Here's a guide to the evolution of Pokémon.
Stopping Evolution
Before I get into the different ways of having your Pokémon evolve, I'll tell you how to stop them from evolving. When you get to the fancy screen of evolving, simply press the B button, which is on all of the game systems for the RPG games. When you do, it won't evolve. But, your Pokémon will want to evolve again the next time it levels up. So, either press the B button again, or let your Pokémon turn into it's beast evolution self!
Evolution by Leveling Up
This is most likely the most common way of having your Pokémon evolve. Lets take a Torchic for example. Torchic is set to evolve into Combusken at level 16, so when you help him reach this level, Torchic will become a Combusken. A Combusken will evolve into Blaziken at level 36, so again, when he reaches this level, it will evolve. Easy enough? Yeah, but every Pokémon has it's own set level to evolve at, so you'd better check up on that. Ontop of that, some Pokémon just don't evolve altogether.
Evolution by Stones
Certain Pokémon evolve by using stones on them. You just need to give them the right stone, and they'll evolve on spot. Pokémon that evolve by using stones include Raichu and Clefable. Pokémon who evolve this way won't evolve with any stone, you need the right one. To get Raichu, you need a ThunderStone, and with Clefable, you need a MoonStone. Sometimes the Pokémon you want to evolve may have different evolutions, so it's up to you to pick evolution you want, by the use of the right stone. This is the case with the three Kanto Eeveelutions.
Evolution by Trade
To make some Pokémon evolve, you need to trade them. It's simple. Just get a friend, trade the Pokémon you know will evolve over trade over to you friend, it will evolve, trade it back, and you got it! Examples of Pokémon who only evolve over this method are Alakazam and Machamp.
Evolution by Happiness
I don't know much about this one, as I've never tried it. But, some Pokémon only evolve when they reach a certain happiness level, and then they evolve. I don't know how to get them happy... So don't ask.
Evolution by Other Methods
There's always those weird ones... Isn't there? Yeah... So some Pokémon need to have a certain move when leveling up to evolve. Some need to get happy at night time. Some need to level up near a stupid Ice Rock... Some need to hold an item while trading. That's just the world of Pokémon!