Colony ships are used to colonize a new planet. They turn into a Planetary Command Center upon landing, and they carry all the necessary people and supplies to start a colony. When you send a colony ship the necessary happiness is subtracted from the planet.
When a colony ship is sent to a planet which requires a terraforming platform and/or gas giant settlement platform, it creates a planetary command center (PCC) and the planet starts at -1 available plots.
When you send a Colony Ship (or Space Station Hull) the current Next Colony Cost amount of happiness is deducted from the sending colony and the Next Colony Cost is multiplied by ( 2.75 - 0.01 x Growth Affinity ). If the Colony Ship (or Space Station Hull) is killed (usually by a SAW in the same system as the target planet or a Fighter in orbit around the target planet) then you will not get the happiness cost back, but your Next Colony Cost will be divided by the multiplier (putting it back where it was before you sent the ship).
Note: When selecting a location for a new colony you should consider checking the Population by zone page.