New to Beyond the Sword is the ability to make colonies of your empire. To do this you must have at least three cities on a landmass other than your home continent (where the capital is). Opening up the domestic screen will then open up this option.
Note that you can only form one colony on each continent; any other liberation on a continent which has a colony of yours already will immediately be given over to the preexisting colony.
After creating a colony it will immediately become a vassal (even if you do not have Feudalism yet); you will get a +10 relations boost because they are your colony, and you will be able to demand any resource from them that you do not have.
They will start with any of the cities you have chosen to release (and you can liberate more to them if they are on the same continent). They will also be given automatic defense units, and will start with every technology you had when they were turned into a colony.
Colony leaders are chosen by the most similar cultures. For example: Britain makes a colony. It is lead by the Americans.
If a colony is mistreated, it may break off and form a completely independent empire, in which case it is time to "send in the troops."