The colony ship is the third basic ship type, and as with the other two it is essential to your winning the game. Colony ships are how you carry your civilization out from your Home World and establish outposts on other planets. These planets become part of your empire and add economic resources as they grow.
Colony ships deposit population on the new planet equal to the square of their current BR. That is, if your colony ship's BR score is below its maximum, then the resulting population will also be below maximum levels. This might, for example, result from colonizing after combat if of you have overbuilt to the point of driving your maintenance levels below 1.0. Timing can therefore be important. Either time an combat/overbuilds so that you are not colonizing during that turn, or simply set the colony to standby and colonize the next turn at a lower amount.
As with science ships, there is debate over when and how to colonize. Early colonization will lead to more planets, but if you wait for higher BR you will plant larger populations. I personally prefer early colonization, though, because that gives you the option of early pop tricking.
In the middle and late game it is important to colonize with as large of a colony ship as possible. Yes, they will cost more to build and maintain, but this will be more than made up for by the population it plants. If you use a BR-1 colony ship it will take several turns for the planet to reach the population you would have achieved if you had used a higher BR colony shop. This can be critical if you need the economy or if you need that planet to become a builder as fast as possible.
It is advisable for an attacking fleet to bring along at least one colony ship so that you can immediately colonize the planet after nuking it. This is especially important if you have high BR and can make the planet a builder quickly (remember, a BR-7 colony ship will create a population of 49) or if you have a stargate that can be used to immediately gate more ships to that planet.
Note that if you have multiple colony ships in the same planet, only one of them will be able to colonize the planet — their affects are not cumulative. But, once one ship colonizes the other simply reverts to standby status, so you didn't lose it. It is therefore advisable to go ahead and set all colony ships staying in that system to colonize, that way if the enemy attacks at least one of them might survive to colonize the planet.