Evolution is when Pokémon transform to other species, exclusive to their evolution lines. There are some ways by which it happens.
Ways of evolution
Sometimes when a Pokémon levels up, it evolves. To grow levels, your Pokémon must get a certain amount of experience points. The experience it gains causes it to gradually mature until it finally wants to evolve.
To cancel Pokémon's evolution until the next level-up, repeatedly tap the button during evolution, or have the Pokémon hold an Everstone. This is useful for Pokémon to learn a move that its evolution won't or only learns much later. Preventing a Pokémon to evolve will affect its stat growth only while it remains unevolved. Once it evolves, its stats will increase to the level it would have had as if it had evolved at its proper level.
Leveling-up with high friendship
Some Pokémon only evolve when leveling up if their friendship level is high, past a certain amount. This is a more modern way of Pokémon evolution where Pokémon need to have bonded with their trainer. It allows Pokémon to evolve at unusually early levels. An example is Chansey evolving into Blissey. Daisy, from Pallet Town, roughly tells players how friendly their Pokémon are.
Trading while holding an item
Some Pokémon evolve when traded while holding an item, which is also another more modern method of evolution. An example is Scyther, which evolves into Scizor if it's traded while holding a Metal Coat.
Leaf, Sun, Moon, Fire, Thunder, and Water stones also evolve Pokémon. An example is using a Water Stone on Staryu, so that it evolves into Starmie. Evolution stones allow Pokémon to evolve at unusually early levels.
Other ways of evolution belong to other Pokémon games and it's important to know that they will not work in the game that you are playing, which is FireRed and LeafGreen. Make sure that the evolution method belongs to the game being played but also that the evolved Pokémon exists in these games, as some evolved Pokémon may have been created only after these games were already made, thus being impossible to appear on it.
Branched evolution is a trait of Pokémon that can evolve into one of several possibilities. Eevee is the flagship of this method. It can evolve with at least three different evolutionary stones, each resulting in a different species.
Obtaining Pokémon evolution lines backwards
Some Pokémon can only be obtained already evolved. To get their unevolved forms, breeding and holding items is required. Breeding is enabled towards the end of the game, but is left largely unexplained in-game, confusingly. By breeding, Pokémon essentially mate and produce an offspring. The offspring might be unevolved, especially if the Pokémon breed while holding certain items.