Pokémon FireRed and LeafGreen/Evolution

Evolution is when Pokémon change forms. There are many different ways this happens:

Leveling-up
Sometimes when a Pokémon levels up it evolves. To grow levels your Pokémon must get a certain amount of experience points (exp). Obviously the only way to do that is to have it participate in battle (EXP share also works).

Trading
Using a GBA link cable you can trade certain Pokémon to other trainers. Some Pokémon evolve when this happens.

Items
When some Pokémon are given items they evolve.

Trading with Items
Rare items are known to evolve Pokémon but only if they are traded with the item

Stones
Leaf, Sun, Moon, Fire, Thunder, and Water Stones also evolve Pokémon.

Random
Random evolution occurs when one Pokémon may turn into one of two different Pokémon.

Taming
When Pokémon are happy enough they might evolve.

Day/Night
Some Pokémon evolve whether it is day or night, this does not work with FireRed/LeafGreen because they don't have an in-game clock so you must trade the Pokémon over to Ruby/Sapphire.