Many of the non-muslims in Egypt are Coptic Christians, a branch of the Christian church that is based on the teachings of St. Mark, who came to Egypt in 50 CE. They are related to the Greek Orthodox church, but they have their own leadership and administrative structure.
Copt, or Coptic, is merely a corruption fo the Greek word for Egyptian - Aegyptios.
Christianity spread quickly in Egypt, stemming from a community in Alexandria in the first century. Similiarities between the Christian religion and the beliefs o the Egyptians (including remarkable parallels between the stories of Christ and Osiris) made it easier for Christianity to gain converts and a strong foothold in Egypt.
The Coptic quarter of Cairo is the "Old City", and contains a number of churches and monasteries.
Coptic is also the term used to describe the written form of the language, one of the latest forms of written Egyptian.