Ancient Egyptians conceived of a natural force, called heka, which was created by Re before time as a resource for humans to use to ward off bad things. I suppose in a way it's like static electricity, in that some objects will hold it, and a user of heka can direct it. To speak a spell is to have heka in one's mouth. Powerful magical items hold and direct lots of heka; some things have more intrinsic heka than others: like the king, graveyards, books, gold, names, precious stones.
Heka was also used to combat the forces of chaos -- the desert, storms, sickness, dangerous animals, and foreigners -- in order to uphold "maat," or truth and order. (The imagery of foreigners in ancient Egyptian hieroglyphs, foot ware, religious iconography, and in magical texts is the xenophobic elephant in the pyramid of Egyptian studies.) Heka could form a protective shield encircling the magician, or could in turn encircle and bind the forces of chaos.
Ancient Egyptian spells involve ritual actions or gestures, special or exotic focus objects, and written or spoken words -- especially names. So if you're writing a magic scene set in ancient Egypt, your spell-caster is going to be waving around an ankh (at least) while using as many True Names as possible. Spells to subdue an enemy might involve breaking pots with a person's name written on them, or melting wax images of them. In some rituals, a magician/priest has the goddess Maat painted with white paint upon his tongue so that his words are true and pure.
Colors had magical correspondences. Green was associated with plants, and therefore flourishing growth. Blue (like lapis) was associated with heavenly powers. Black was a powerful color (maybe associated with the fertile black soil?). Red was a color of power, but also chaos, associated with the chaotic god Seth. Magical scrolls might have magical names or chaotic powers written in red ink (otherwise, they mostly used black).
Scholars like to spend a lot of time writing about the boundaries between ancient Egyptian religion, magic, and medicine. This is because the boundaries between the three are blurred. Ancient Egyptian (Early- and Middle-Kingdom, at least) spells tend to begin with a story about the gods as a kind of "this is the way the universe works" starting statement, and then has the spell caster identify with one of the god-protagonists in the story. Also, magic spells were part of a non-exclusive toolkit -- along with prayers and mundane actions -- for healing or averting ill fortune.
Scholars also like to talk about how ancient Egyptian magic is different from the modern (-ish) western concept of sympathetic magic as put forward by Sir James Frazer. I'm not sure there are that many differences, since ancient Egyptian magic operates with the Law of the Macrocosm and Microcosm, the Law of True Names, the the Law of Contagion. On the other hand, I'm not sure how to classify a spell that requires one write write the spell onto one's body and then lick it off for it to work -- consuming or otherwise taking the spell into one's mouth was a way for illiterate folks to activate the potency of a spell.
Simplifying things greatly, ancient Egyptian magicians came in three flavors: the king as high-priest of the nation casting spells to uphold order (maat), a scribe-priest associated with a temple's scriptorium (or House of Life) who might cast healing spells or compile magical guidebooks for navigating the afterlife, and common folks who used charms, talismans, and magical gestures during times of crisis (like birth or death) to manipulate health and luck (and hippos and crocodiles) or avert the evil eye.
Although there was a mention of foreign (Nubian) women and their terrible spells, most magicians in ancient Egypt were male priests working out of a temple. There might be a sample bias here, as temple priests were more likely to leave a record of spell (or medical triage) books. Being a priest was a part time job, and when they were off of temple duty, they were typically doctors or scribes. As time went on, the priesthood became hereditary. So if you're going to write an ancient Egyptian magician, they're going to be part of a literate elite, or connected with the royal court.
Ancient Egyptians made heavy use of amulets, like the ankh (for life); others include the djed pillar (for stability), the shen (for protection), the scarab (for regeneration), and the wedjat eye (for wholeness and protection). Amulets could be as simple as a knotted thread, or a magical word or symbol written onto a piece of cloth and put into a small bag, or even a tattoo.
Finally, ancient Egyptian magic was concerned with helping folks attain a good afterlife. Afterworld spells can be attested throughout the Egyptian kingdoms, starting with the Pyramid Texts (~2353 BCE, and which were reserved for the king), to the Coffin Texts (~2100 BCE), to the Book of Gates (~1500 BCE) , The Book of Going Forth By Day (~1550 BCE, available to the upper classes), The Book of the Hidden Chamber, The Book of Adoring Re in the West (~1425), and other Netherworld Texts. These contained spells and rituals a person would need to recite in order to navigate the perils of the netherworld or Duat and unite their ka (or spirit) with their ba (or soul) -- much in the same way Re the sun god was thought to unite with Osiris the mummiform god of the underworld. The Book of the Heavenly Cow (~1341) appears to be a collection of stories featuring gods and sorcerers.
Once we get to around 300 BCE, Egyptian magic starts to look more familiar. For one thing, it seems to be more about curses and bindings and less about protection, healing, and the afterlife. The gods become more syncretic. The spells begin to become more abbreviated and cryptic. During this time we start to see gods like Abraxius, and magical anagram-like words, like abracadabra make their appearance.
I suppose if I were going to write about an ancient Egyptian magician, I'd do an alternate history magician. They would need to be able to read and write. They'd need to have good observational skills in order to detect and move heka. They'd need to be versed in the creation myths of their society in order to make use of the Law of Macro-and-Microcosm. They'd be a boy-scout type concerned keeping the system running orderly. I'm split on what gender to make them, although writing a non-traditional / non-male would be give them a social hurdle to get over. Or maybe I'd make them a foreign magician trying to work within their adopted land's system (more opportunities for conflict there) -- maybe they could be a lover or spouse of a native. I'd probably make my magician a mystery solver -- so I guess a police procedural or Brother Cadfael type of character.
. . . or . . .
maybe I could make them a kind of shabti figure (a kind of Egyptian golem). . . doing work for a magician. . .
. . . or . . .
maybe this school-teacher / anthropologist is digging in modern Egypt, and she finds this box from the time of Queen Hatshepsut, and inside the box is an amulet of Isis, and....