The term moral obligation has a number of meanings in moral philosophy, in religion, and in layman's terms. Generally speaking, when someone says of an act that it is a "moral obligation," they refer to a belief that the act is one prescribed by their set of values.
Moral philosophers differ as to the origin of moral obligation, and whether such obligations are external to the agent (that is, are, in some sense, objective and applicable to all agents) or are internal (that is, are based on the agent's personal desires, upbringing, conscience, and so on).
Obligation being a set code by which a person is to follow. (Obligations) can be found by an individual's peers that set a code that may go against the individual's own desires. The individual will express their morality by the person following the set code(s) through seeing it as good to appease society.