The key to packing for a trip to Egypt is to focus on lightweight and practical items for daytime sightseeing. Cottons, linens, and moisture-wicking fabrics make the most sensible choices for the heat. Egypt is an Islamic country, albeit a more open society than some in the Middle East, but attitudes toward dress are still more conservative than in the United States, particularly with regard to women's attire. The clientele in Cairo's upmarket hotels and nightclubs tend to dress up. Resort towns on the Red Sea are the exception; foreign women are more or less free to dress how they want.
It's important that all travelers—but particularly women—not expose too much flesh. Pack T-shirts with sleeves that end between shoulder and elbow rather than tank tops or those with spaghetti straps. Many women wear light scarves to cover their necks and shoulders; these can be bought cheaply on the street or in bazaars. Long shorts and Capri pants are fine for women, but full-length pants are better. Skirts should be at least knee length. Short shorts and short skirts will cause stares; moreover, to visit churches and mosques in Egypt women must have shoulders and knees covered. In mosques you'll also need to cover your hair, so if you don't want to use a scarf supplied by the mosque, carry your own lightweight scarf.
For men, long shorts are acceptable when you're traveling on tours, but full-length lightweight pants are preferable and are especially recommended in Cairo and in the desert, where they offer more protection from the sun. Regular T-shirts are fine, but lightweight collared shirts help protect your neck and arms from the sun better. Only a few hotels require a jacket and tie (most notably the Sofitel Winter Palace in Luxor), but men will be expected to wear long pants, collared shirts, and shoes (not sandals) in the evenings.
If you travel in winter, pack a fleece or a jacket for the cold evening air. This is especially true if you intend to overnight in the desert.
Beachwear is appropriate only around the pool or at upscale beaches. You may find visitors of other nationalities scantily dressed in hotel lobbies and even around town, but this is against local sensibilities.
Must packs: comfortable shoes, because you'll be walking a lot and climbing up and down rickety or badly set stairs into tombs; a hat, because the sun is hot at all times of year; sunglasses, because temple facades and rock faces are extremely bright in the daylight; and sunscreen to protect any exposed skin. Pharmacies in Egypt are well stocked. Don't bother bringing expensive prescription medicine for intestinal upsets. Instead, buy Antinal, a locally available intestinal antibiotic that is commonly used to treat diarrhea, when you arrive in Egypt, and take a couple the moment you feel any problems. It's cheap (about $1.50), available in every pharmacy, and very effective. In more serious cases, a doctor or pharmacist can prescribe a stronger antibiotic, which will cost much less than at home.
Extra stuff that will be helpful includes: a small flashlight for visiting dimly lit tombs and temples; lightweight binoculars, to allow you a clearer view of monumental temple facades, and for bird-watching on the Nile; antibacterial gel, so that you can clean your hands before eating no matter where you are.
Except in the Western Desert, you can buy almost anything you need in the cities and main towns, from baby formula to feminine hygiene products and contact lens supplies, but prices may be more expensive than at home.