Happiness (also known as Friendship) is how happy the Pokemon is. There are many ways in many games that can increase a Pokemon's happiness. Your party Pokemon are probably the most happy of all the Pokemon you own, since they have gone through countless battles with you. Through constant battling, your Pokemon can gain a lot of happiness.
You can check here for all the methods of how to increase your Pokemon's happiness.
You can also check here to see who can rate your friendship in each game. Then you can go down the page to see what they say when.
If I remember correctly, max happiness occurs when it reaches 220. (I may be wrong though)
Hope I helped. :)
Source: Above links