What is considered an "unsecured priority claim"?

Unsecured priority debt as falling in between secured and general unsecured.  The creditor does not have any special right to any of your property (such as a mortgagee would in your home), but the bankruptcy laws give them special rights so that they have a priority over other unsecured debt (such as credit cards).

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, overpaid benefits, and most student loans. 

In a Chapter 13 case, priority debts have to be paid in full, unless the creditor agrees to only take partial payment. You do not have to pay interest on priority claims.

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.