Project Xanadu® is the original hypertext system project, established in 1960. What qualifies that statement? The Project Xanadu home page, of course!
The word hypertext was coined by Ted Nelson in 1965 and Xanadu was the name he gave to the global hypertext system that he envisioned and designed over subsequent decades. The name first appeared in 1966, and is a reference to the "lofty pleasure dome" in Samuel Taylor Coleridge's "Kubla Khan" (the Alph project, likewise, takes its name from the sacred underground river in the same poem).
Two historical documents can shed better light on Nelson's ideas for Xanadu than what I can write here. To get The Big Idea, have a look at pages 56 and 57 from Nelson's 1974 book Dream Machines:
And for a detailed chronicle of the Xanadu adventure up to 1987, I strongly recommend reading Literary Machines 87.1:
I think a great one-liner for Xanadu which dates from after the spread of the World Wide Web is this one:
"an online world-wide anarchic populist hypertext network in which anyone may publish with two-way link overlays creatable by anyone."
I don't have a definite source for that one, but it's referenced in the alt.hypertext FAQ as coming from Nelson in 2003.