Origins
The English word 'Egyptian' descends through Latin 'Aegyptius' from Greek 'Aigýptios' (Αἰγύπτιος), the adjective formed from 'Aígyptos' (Αἴγυπτος), the Greek name for Egypt. This Greek name appears as early as the Mycenaean period (the Linear B form 'a-ku-pi-ti-jo' has been tentatively read) and is firmly established in Homer, where 'Aigyptos' refers both to the Nile River and to the land through which it flows.
The most widely accepted etymology traces 'Aígyptos' to the Egyptian phrase 'Ḥwt-kꜣ-Ptḥ' (conventionally vocalized as 'Hut-ka-Ptah'), meaning 'House of the ka (soul/spirit) of Ptah.' This was the name of the great temple complex of the creator god Ptah in Memphis, the ancient capital situated at the apex of the Nile Delta. Memphis was the first major Egyptian city encountered by travelers arriving from the Mediterranean, and it is entirely plausible that foreigners extended the temple's name to the entire country — a process paralleled elsewhere in history (the name 'India' derives from the Indus River, which is only the country's northwestern boundary).
The Egyptians themselves never used any form of 'Aígyptos' for their country. Their own name was 'Kmt' (usually vocalized as 'Kemet'), meaning 'the Black Land' — a reference to the dark, fertile alluvial soil deposited by the annual Nile inundation, in contrast to 'Dšrt' ('Deshret,' 'the Red Land'), the barren desert on either side. This black-soil/red-desert dichotomy was fundamental to Egyptian self-conception: Egypt was the narrow strip of life along the Nile, bounded by the lifeless desert.
Greek Origins
The Arabic name for Egypt, 'Miṣr' (مِصر), has yet another etymology, tracing to the Akkadian 'Miṣru' (boundary, border, territory) and possibly to an ancient Semitic name for the country. 'Miṣr' appears in the Hebrew Bible as 'Mitsrayim' (מִצְרַיִם), the dual form perhaps reflecting the ancient division between Upper and Lower Egypt. Thus Egypt has at least three completely independent name traditions: the Greek/European (from Hut-ka-Ptah), the Semitic (Miṣr/Mitsrayim), and the native Egyptian (Kemet).
The Egyptian language itself has the longest documented history of any human language, spanning over four thousand years from the earliest hieroglyphic inscriptions (c. 3200 BCE) to the final extinction of Coptic as a spoken language in the seventeenth century CE. Linguists divide this immense timespan into five phases: Old Egyptian (c. 3200–2100 BCE), Middle Egyptian (c. 2100–1600 BCE, considered the classical literary language), Late Egyptian (c. 1600–700 BCE), Demotic (c. 700 BCE–500 CE), and Coptic (c. 200–1700 CE). Coptic, written in a modified Greek alphabet, is the final stage of the Egyptian language and survives today as the liturgical language of the Coptic Orthodox Church.
Egyptian belongs to the Afroasiatic language family (formerly called Hamito-Semitic), making it a distant relative of the Semitic languages (Arabic, Hebrew), the Berber languages, the Cushitic languages (Somali, Oromo), and the Chadic languages (Hausa). Within Afroasiatic, Egyptian forms its own independent branch — it is not Semitic, despite sharing some features with Semitic languages due to contact and common ancestry.
Legacy
The decipherment of Egyptian hieroglyphs by Jean-François Champollion in 1822, using the Rosetta Stone as his key, was one of the great intellectual achievements of the nineteenth century. Before Champollion, the hieroglyphs had been unreadable for over 1,400 years, since the last known hieroglyphic inscription was carved at the Temple of Philae in 394 CE. Champollion's breakthrough opened the entire civilization of ancient Egypt to modern scholarship and made 'Egyptian' not just a language name but a gateway to one of humanity's foundational cultures.