The evolution system on Pokémon GO is a whole new concept. No filled PC boxes here, but free candy for every capture! So, to evolve a Pokémon, you need a certain amount of candy that the Pokémon gives out when you catch it.
To evolve your Rattata into a Raticate for example, you need to catch around 10 Rattata to obtain 25 candies and evolve it.
Candies can also be used to Pover Up your Pokémon, i.e. boost its CPs. If you run out of candy, you can get some from Professor Willow by 'transferring' your surplus Pokémon to him.
There are also Pokémon that can only evolve through items. These were added with the arrival of Gen 2. Depends of the Pokémon, you'll need 50 or 100 candies + 1 item to evolve it.
Trade Evolutiion
Finally, with the arrival of Gen 5, it's now possible to evolve some Pokémon by trading, just like in the original games.