What is TeX, and why does it still matter?
TeX is a typesetting program: you write plain-text markup describing a document, and TeX turns it into finished pages — correct hyphenation, even spacing, mathematics set the way a human typesetter would set it. Most people today reach it through LaTeX, a set of macros built on top that makes the plain TeX language easier to write in.
Why it exists
In the late 1970s, Donald Knuth — a computer scientist, not a typesetter — was revising the second volume of The Art of Computer Programming. The publisher's new digital typesetting system came back with proofs so much worse than the first edition that he balked at them. Around the same time he saw a book that had been typeset well by computer, and made the connection: typesetting is just arranging ink and no-ink correctly. He thought a proper solution would take about six months.
It took closer to ten years, with help along the way from typeface designers including Hermann Zapf, Chuck Bigelow, Kris Holmes, and Matthew Carter. The result was two programs: TeX for laying out text and mathematics, and Metafont for describing the fonts themselves — both released for no fee, with source available to anyone, at a time when the commercial alternatives were expensive, proprietary, and often machine-specific.
Why it lasted
The feature that mattered most in the long run wasn't the typography, it was the promise underneath it: the same input file produces the same output — same line breaks, same page breaks — on any machine, forever. That's unusual. It's why a thesis or a paper written in LaTeX in 1990 will still typeset correctly today, and why the format never became beholden to any one company's software surviving.
TeX itself has been declared frozen for years, deliberately: past a certain point, stability outranked new features. The active development happens around it — in LaTeX, in the fonts, and in the free TeX Live distribution TUG helps coordinate.