Most evolutions happen through level. most first evolutions happen at around lvl 16. Such as Charmander to Charmeleon. But after the first evolution most Pokemon vary in lvl due to power and strength. Such as Ivysaur to Venusaur who evolve at level 32 rather than the usual level 36 that the first starters evolve at. If your pokemon evolves through level all you need to do is train them up or use rare candy to level them. Once they hit the right level they will evolve. The three original starters are great examples of this.
Some Pokemon evolve from Happiness, Happiness can be increased by winning battles with your Pokemon or keeping them out while you walk. As well as these generic ways of increasing your happiness you can also use a held item called a soothe bell. This item boosts the friendship of your Pokemon. Most Pokemon who evolve through Friendship do so at 255 happiness. Such as Riolu to Lucario. But with most the Friendship pokemon they need to be levelled during a certain time of day with the correct happiness in order to evolve. Such as Riolu who needs to be levelled during day time.
Many Pokemon evolve by using a item on them such as a evolution stone. Each stone is connected to a Pokemon type. For example: fire stone, leaf stone and water stone. The most well known example of a pokemon who evolves with an item is Eevee. Almost every single Eeveelution evolves from a stone.
There are many other Items that evolve Pokemon including Trading items or Held items you level your Pokemon up with. For example: Magmarizer Evolves Magmar into Magmortar when traded with the item on the Pokemon And Razor Claw evolves Golbat into Crowbat at night when levelled up once. The Magmarizer and Electrolizer are shown bellow.
Finally some pokemon can only evolve through Trade, For example one of these Pokemon is Machamp. Machamp Can only be obtained by trading a Machoke With another player.
5) Evolution Prevention.
Although you may think you always want your Pokemon to evolve that may not be the case. Your Pokemon may learn a move as a lower stage Pokemon but after its meant to evolve. So you use an item call a Evertone. This item prevents any Pokemon for evolving. In Pixemon you can also disable the levelling on your Pokemon in the Summary tab. This means you won’t Level and you won’t Evolve.