The weather in Egypt is hot but the nights can be cool, so it is important to bring a few layers, perhaps a sweater that you can wear in the evening. You may also want to bring a pair of jeans for the cooler nights. In the day time, bring shorts and t-shirts in lightweight fabrics. If you plan to spend time by the beach or by the pool then you will also want to bring a bathing suit as well. You should also remember that women are expected to dress more conservatively. This means that you should avoid wearing short skirts. You should also avoid wearing barebacked dresses or midriff tops.
The sun in Egypt is probably hotter than you're used to, so it is essential to bring sun block in a high SPF such as SPF 30. If you'll be going in the water, generally gel sun blocks are better for in the water while cream-based ones work better out of the water. Make sure to pack some sunburn relief products, just in case. In addition, make sure to bring sunglasses to protect your eyes. You will also want to bring a wide brimmed hat to protect your head. Finally, consider packing a canteen to prevent dehydration.
In addition to sun protection it's a good idea to include some insect repellent. You will also want to bring some tissues or moist towelettes as it can be very dusty, especially in Cairo. Also be sure to pack some Imodium or similar type of medication, just in case.
What extra gear you bring will vary depending on the types of activities you enjoy participating in. In general, good walking shoes are a must. Bring two pairs, if possible, as they will get very dusty and dirty. Also, it is a good idea to bring a good flashlight with you if you plan to see inside the Egyptian tombs. Camera and spare batteries are also essential. If you enjoy bird watching, consider packing a compact set of binoculars so you can appreciate some of the Egyptian wildlife. For those that will be scuba diving by the Red Sea, remember to pack accordingly. There are numerous dive centers in Egypt, but some scuba divers prefer to pack their own gear. If you won't be scuba diving but are spending time by the Red Sea, definitely consider packing a snorkel, mask and fins so that you can do some snorkeling while you are there.