When people use the term "high and mighty" as an idiom in everyday speech, they are usually referring to unwarranted arrogance in an individual. For example, a person with wealth and power who acted as if he was entitled to better treatment than others might be described as acting like he is high and mighty. In theory, the term could also be used in a complimentary way to describe how powerful someone was, but in actual use, it is almost always meant to insult or make people reconsider their relative status.
Over the centuries, people born with royal blood have often been considered "high born." The public often thought of them as having higher status than regular people, and some cultures actually attributed this to an association with divinity. Experts believe that the idiom "high and mighty" developed as a way to mock rulers for taking too much advantage of their status. For example, if a ruler was eating expensive food while his people starved, that could be cause to mock the individual using the idiom. Over time, the application of the term evolved, with people using it to describe behaviors rather than a reference to a specific kind of birth background.
One of the main reasons people use this idiom is to describe arrogant behavior, and people who behave with a sense of entitlement and ignore the needs of others might be mocked using the term. When people use this idiom in a confrontational manner, they might intend to make an individual reconsider his arrogance, bringing the person back down to Earth, but it is also often used behind a person's back. For example, if a manager was acting high and mighty, the workers under him might use the term during conversations about him while he wasn't present.
Another common use of the idiom "high and mighty" is when someone oversteps his bounds of authority. For example, if a person who is equal to others in a group started bossing people around, that person might anger his co-workers, and they might let him know right away that he should stop acting so high and mighty. This sort of use of the term could work in any situation where the people are of relatively equal status, including many kinds of personal relationships.
Sometimes the term is used with lot of intentional sarcasm. For example, someone at a party might say, "Tom would be here, but he's so high and mighty now that he doesn't want to hang out with his old friends anymore." In theory, the term could also be used in a complimentary fashion to describe someone's ascendancy to a greater status, but that is basically unheard of.