When you "pay something off", it means that you completely pay for it. This is mostly used to talk about bill and loan payments. When you "pay off" a loan, it means that you have completely paid all of the money you borrowed, plus the loan's interest.
To "pay off" a credit card means to pay off all of the money that you spent on that credit card for the month.
Credit cards in the U.S. have a minimum payment that you have to pay each month. However, if you only pay the minimum, you will be charged interest for whatever amount hasn't been paid. If you "pay off" your credit card each month by the deadline, you won't be charged for interest.