A written and legally binding promise to do something, or to refrain from doing something.
Failing to fulfil a covenant, regardless of circumstances, leaves you liable to be sued.
For example, when you apply for a mortgage loan, you will sign a covenant promising to pay your property taxes, promising to not destroy the house, and promising to repay the mortgage loan in full.
Some people believe that if their home is sold under power of sale, they can walk away from the mortgage loan. This is not true. Because they signed a covenant to repay the mortgage loan, they are obligated to do so regardless of the circumstances. Though the proceeds of the power of sale are applied to the outstanding loan amount, the person is still on the hook for the remaining amount because of their covenant.