For the post count rank, see the FAQ
A captain in the Total War series is a man who leads an army when no general is present. He has a command rating, usually quite low, but no traits or ancillaries.
Captains can be adopted into the family through the "man of the hour" event. If a captain gains enough good victories, sometimes only 1, you will be presented with the option to allow him into the family as an adopted son. This event is more likely to take place if your faction's family is smaller than expected for the amount of territory you hold.