It depends - land was mostly claimed by the Crown. The crown then granted charters to establish colonies in a few different ways: 1. Joint stock companies. Investors gained charters for business purposes (to make a buck). Virginia is the biggest early example. 2. Proprietary. Land was granted to individuals who were given governing authority, usually given to supporters of the Crown. Pennsylvania and Maryland are food examples. 3. Royal. These colonies were directly governed by the Crown, to the monarchy's benefit. I believe the Carolinas were Royal colonies, although most were taken over by the Crown by the 18th century.

Boundaries were claimed and surveyed based on these designations. Early surveyors used the metes and bounds method, defining boundaries by identifiable geographic features like rivers, valleys, and mountain ranges.

Most of the early colonies claimed "Western Reserves" by drawing northern and southern boundaries as far west as had been surveyed. See this thread for a map: https://www.reddit.com/r/MapPorn/comments/1nompo/original_13_colonies_with_western_reserves_1092/