The happiness of a Pokemon is the bond and emotions the Pokemon has for its trainer. Some Pokemon need a certain amount of happiness to evolve, such as riolu and munchlax. Others use happiness to make there moves stronger(can only find return). There are many ways to increase your Pokemons happiness. You can have someone give it a massage, battle with it a lot, giving it drinks like protean and zinc, and, most recently, play with it in Pokemon amie(I know this is a 5th gen question, but I just had to add this). But be warned, there are ways to decrease your Pokemon's happiness, though I only know making lose in battle. The reason you should raise your Pokemon's happiness is to evolve it, make it stronger, and just to increase your bond to it.
Have fun ;)