The word 'obligation' refers to the idea of being bound by a duty. Legal obligations create legally enforceable duties. For example, a legal obligation to repay a loan means that the borrower has a legal duty to repay the loan.
An obligation can be discharged by performing the duty in question, e.g. by repaying a loan. If the obligation is not discharged, the person with the corresponding right to be paid may bring a legal action to enforce the obligation.