This is the ordinary country adjective, and it stretches over everything connected with Egypt — a person, a passport, a meal, a temple wall, a style of drawing. English writes it with a capital letter because it comes from the name of the country. One thing worth noticing: because Egypt's history is so long, the same adjective covers both the ancient world of pyramids and the modern country where people live and work today, and the situation tells you which one is meant.

This is the same word used as a noun, naming the person instead of describing them. It behaves like an ordinary countable noun: an Egyptian, two Egyptians, many Egyptians. Notice the little article difference that trips learners up — you say 'She is Egyptian' with the adjective and no article, or 'She is an Egyptian' with the noun, and the first is far more usual in everyday speech. The capital letter stays in both.