Happiness reflects your island's standard of living; and determines how happy citizens are overall. It is controlled by many factors, such as the quality of the citizen's housing and job, how safe from crime and oppression the citizen feels they are, and how much they respect the current government.
Keeping citizens happy is important as unhappy citizens are likely to leave Tropico, vote against the current presidente in elections, take part in protests and uprisings, and even join the rebels.