In all of the Pokemon games, almost all of the different types of Pokemon evolve into at least one or two other types. Evolution can increase your Pokemon's strength and speed. It can also give your Pokemon new attacks or skills. Each type of Pokemon evolves in a different way, and may require different things in order to make the transition. There are really only four ways to make your Pokemon evolve. Which way will work for you depends on which type of Pokemon you want to evolve.
Use your Pokemon frequently in battles so it gains Experience Points. Most Pokemon evolve when they reach a certain Experience Level. For example, Abra will evolve into Kadabra when it reaches Level 16. Each new level is reached by gaining Experience Points. Experience Points are given every time a Pokemon wins or helps to win a battle. In order to gain the points, the Pokemon can't faint during the battle. One easy way to help a Pokemon gain Experience Points is to start the Pokemon in a battle, then quickly switch it with a stronger Pokemon so that the Pokemon you hope to evolve is not in danger of fainting. Both Pokemon you use will gain points.
Use items such as Rare Candy on your Pokemon. These items can raise your Pokemon's Experience Level, making it evolve quicker.
Use an item, such as a stone, to force your Pokemon to evolve. Some Pokemon will only evolve when an item is used. For example, Pikachu evolves into Raichu when the Thunder Stone is used. Pokemon that evolve this way can be forced to evolve at any level.
Keep your Pokemon happy. Some Pokemon, such as Togepi and Golbat, only evolve when they're happy. Keep the Pokemon in your party, never putting them in storage or trading them to keep them happy. Use beneficial items, such as HP Up, on the Pokemon. Give them items, such as Berries, to hold. Catching a Pokemon with a Love Ball can also make them happy.
Trade your Pokemon. Some Pokemon, such as Machoke or Haunter, only evolve when they're traded to another player. Other Pokemon need to be holding certain items when they're traded in order to evolve. For example, Onix needs to be traded while holding the Steel Coat, to evolve. Pokemon that evolve when traded can evolve at any level, so it's easy to force an evolution. However, you should never trade a Pokemon that needs to be happy in order to evolve.