Before the American Revolution, American colonies were run by something of a constitutional monarchy. They were ultimately ruled by England which was, of course, a constitutional monarchy and their colonial governments were smaller versions of that government.
American colonies were typically governed by a legislature and a governor. The legislature was typically made up of two houses. The lower of these houses would be elected by the people. The upper house would be appointed by the governor. The governor, in turn, was appointed by the English government (by the king or his ministers).
This means that the colonies were run like constitutional monarchies. There was a governor who was appointed, not elected and there were legislatures that were partly appointed and partly elected.