Happiness is a measure of how much the people are behind you when it comes to colonizing other planets.  Without this backing your people simply aren't motivated enough to explore outside their known world.  There is no cap to the amount of happiness you can accumulate on each planet, and each planets happiness is independent from the others.

Happiness can be affected in several ways:

• Buildings that give you +happiness/hr, such as the University, Entertainment District, Park, Luxury Housing, Capital, and theme Park.
• Adding food to a park or theme park, even before the party is over and you get the party bonus (this can be immensely useful when construction times are very, oh so very slow or booked in advance for days).
• If you have pollution, you lose 1 happiness per point of pollution that isn't sequestered in a Waste Sequestration Well, Ravine, crater, or Interdimensional Rift.
• Building new buildings that use food give you happiness equal to the amount of food used, provided you have at least 4 types of food available (more than 4 types doesn't give more happiness than 4); demolishing buildings subtract happiness based on the food that was used to build them.
• if you have exactly 3 types of food, happiness doesn't change
• If you have less than 3 types of food, then building and upgrading buildings will cause you to lose happiness.
• Having negative plots affects happiness generation as well.

Each planet that you colonize requires a single lump sum of happiness that is subtracted from your current happiness of the planet you launch your colony ship from, and each additional planet takes an increasing amount of happiness.

Negative happiness increases the build time for everything on the planet by 1% for every 1,000 negative happiness.  The only exception to this is glyph buildings which only have longer build times based on the building level.  The Essentia cost of rush completing a building in the Construction Ministry does not increase due to unhappiness levels.

There is also a cap (9,223,372,036,854,775,807) for the maximum amount of happiness (or theorically, UNhappiness) a colony can have.

Here is a table depicting the first 8 colony costs for each growth affinity.