Royal Colonies were colonies that were governed and administered by the English Crown. This stood in contrast to those colonies set up and administered by the Joint Stock Companies and Proprietary Colonies, where the colony was given to a single individual (or a group of individuals) who would then hold all responsibility (including risk) for the success or failure of said colony.
The thing to keep in mind is that colonization was, by its nature, a deeply risky venture. Generally speaking, the first steps of colonization were almost invariably carried out under either Joint Stock Companies or Proprietors (and this does make a certain degree of sense from the perspective of the English government, as it freed it from carrying the risks and costs associated with those ventures). When you look across the history of the colonies, then, what you would see are colonies starting out as either Joint Stock Company or a Proprietary Colonies, only to be later converted into Royal Colonies.
The colonies which were converted to Royal Colonies were as follows: Virginia, New York, Massachusetts, New Hampshire, North Carolina, South Carolina, New Jersey, and Georgia.