Welcome, vdilly!
A private carrier would provide transportation for only one customer, that being the owner of the fleet. Walmart Transportation is a good example.
For-hire carriers fall into two operating authorities: Common Carrier and Contract Carrier. Nowadays there really isn't a big difference, this link describes the difference better than I could:
Trucking: Motor Contract Carrier vs Motor Common Carrier, contract carriers, mc number
A for-hire carrier may get an account to provide dedicated service to a customer but they would still be engaged with other customers. To use Walmart as the example again, when you see a Walmart trailer being pulled by a Walmart tractor, that is a private carrier; when you see Werner, USXpress, Schneider, etc., pulling a Walmart wagon then you are usually witnessing a dedicated account, where those companies have a contract to continually provide service but they also run their other dedicated accounts and irregular service to anyone who calls and needs a truck.
A lot of drivers are drawn to try to get on a dedicated account because it can provide a more structured environment, get you home more often and generally have you going to the same places. In some cases you make more money than if you were running regional or national but in other cases you make less.