Egyptian History (as we think of it) existed for over 3,000 years, so there was a lot of variation in that period.

The Pharaoh was supreme ruler, though many of his roles seem to have been religious. He acted as the intermediary between the gods and mortals, and thus while not technically seen as divine himself, could assume divine status after death.

Egypt was split into several Nomes, each ruled by a Nomarch. This position was likely hereditary. However, much of the day to day ruling in relation to the average person was done by temple administrators. Temples were everywhere in Egypt (which is why Herodotus says the country is the most religious) and were governed by a priestly and scribal subelite. Temples employed a significant number of people, including cooks, artisans and farmers to work the land. They were also vital for storing and distributing food to the workers, meaning a lot of administratory roles were filled by them. Quite how much power the temples held changed a lot, one source from Dendera seems to suggest priestly positions were hereditary, stating ‘The domain of Hathor, lady of Dendara, will be under the authority of [your son], the proper [heir(?)] of the offices of your fathers’ (Source: Frood, Biographical texts from Ramesside Egypt). During the Third Intermediate Period, the Cult of Amun seems to have ruled upper Egypt in it’s own right without a Pharaoh, however when unified Egypt was always governed by the Pharaoh.