I'll assume you mean the Single Proprietor colonies, of which there were only two: Maryland and Georgia.
Maryland, under the direction of Lord Baltimore, started as a haven for Catholics, who were heavily discriminated against ever since the civil wars in England. They hoped to start again, while still remaining under the rule of the King, they could live thousands of miles away and do their own thing, and practice their religion without fear.
Georgia was led by James Oglethorpe, and served primarily as a buffer colony against Spanish Florida, protecting the valuable colonies of the Carolinas from raids in the process. Plantations and cash crops were introduced there, but to a much lesser extent.
Maryland was established in 1644, and Georgia in 1735.
In essence a proprietary colony was one in which one or more people were granted land by the king. Instead of granting land to a charter company, private individuals were given the land. These individuals were not only granted property rights, but also the rights of a Governor administering laws. In America one of the most famous proprietary colonies was the Province of Pennsylvania, which was granted to William Penn in lieu of a $10,000 payment the king owed his family. Many of these proprietary colonies later became states. Virginia, Maine, North and South Carolina, Georgia Maryland as well as the aforementioned Pennsylvania.