What is a priority (or "unsecured priority") claim?
Think of a priority debt as falling in between secured and unsecured. The creditor does not have any special right to any of your property, but the bankruptcy laws give them special rights. There is a long list of possible priority claims, but the most commons ones are child support, taxes, and any debt owed to a governmental agency, like income taxes, property taxes, and overpaid benefits.
In a Chapter 13 case, priority debts have to be paid in full, unless the creditor agrees to only take partial payment. For instance, if you are way behind on child support payments, you will have to make arrangements to make the ongoing payments every month, but Juvenile Court may agree to let you pay only part of the arrearage, as long as you understand that you will still have to pay the rest of it after your Chapter 13 case is over. You do not have to pay interest on priority claims.
In the long run, a Chapter 7 will not have much effect on most priority debts. Those creditors will be temporarily stopped from pursuing you, but not for long. Those debts will not be discharged, and they will still have to be paid in full.