I've long been intrigued by the story behind the first pilot episode filmed for Star Trek. As any good Star Trek fan knows, NBC passed on this pilot -- which starred Jeffrey Hunter and was produced by Desilu -- but ordered a second, this one with William Shatner, and picked up the series for the 1966-1967 season. The original pilot was later edited into a two-part episode called "The Menagerie" which aired as part of Star Trek's first season. I am not aware of any definitive history of "The Cage," just plenty of stories told and retold over the years, but here's a comprehensive analysis by TrekWeb's Bill Williams of the various versions that have been released on Laserdisc, VHS and DVD. And Memory Alpha, the Star Trek Wiki, also has some information.
The way I understand the story, Gene Roddenberry had the only color print of "The Cage" and it was physically cut apart to create "The Menagerie." Roddenberry was left with black and white 16mm work print that he would screen for fans at conventions. On August 7th, 1986 the Museum of Broadcasting (later the Museum of Television and Radio, now the Paley Center for Media) held the first public screening of the pilot; it was a combined version featuring the color scenes from "The Menagerie" and black and white footage from Roddenberry's work print. Here's how an article from Boston Globe explained how "The Cage" came to be:
It was the tight budget that caused the original "Star Trek" pilot to be lost for more than 20 years. "It was made under Desilu, which was a low budget studio," explains Rodenberry, "I was running out of money making Star Trek episodes, and they Desilu asked me if it was possible to work in the first pilot with Kirk and the new crew. If it could be done, we could have two episodes for the price of one. And maybe make it through the year." Rodenberry said it was probably the most difficult job of writing he'd ever done. The result was the two-part episode "The Menagerie," which won the international Hugo Award, given to works of fantasy literature and art. Unfortunately, the only color copy of the first pilot was used as the work print, because Desilu apparently wouldn't pay $200 to make a copy. Rodenberry has since found many of the original color frames and spliced them with black and white footage to create "The Cage" being shown at the museum. 
An article in The New York Times stated that the pilot ran 65 minutes . A 73-minute version was released on video in November of 1986 (it cost $29.98) but it contained an introduction and conclusion from Roddenberry . In October of 1988, a full-color version was broadcast as part of "The Star Trek Saga: From One Generation to the Next," a special hosted by Patrick Stewart. According to The Los Angeles Times, it "was recently found in the Paramount vaults" . Memory Alpha, however, explains that "in 1987, film archivist Bob Furmanek discovered the missing trims from the color 35 mm negative of "The Cage" at a Hollywood film laboratory, and saw that they were returned to Paramount."
The question I'm left with is whether any footage from the original, complete version of "The Cage" is still missing. The TrekWeb article mentioned earlier concludes that at least some frames were lost during the creation of "The Menagerie" but suggests that only roughly 20 seconds are unaccounted for. And some of these missing frames were included in the 1988 broadcast only to be removed when the pilot was released for the first time on DVD in 2001. Thus, Roddenberry's original vision of Star Trek does exist. For a wonderful look at scenes cut from "The Cage" before Roddenberry finalized the pilot as well as behind-the-scenes information, head over to Star Trek History.
2 "A First showing for 'Star Trek' Pilot." New York Times. 22 Jul. 1986: C.18.
3 Martin, Sue. "Turn-Ons and Turn-Offs in Current Home Entertainment Releases." Los Angeles Times. 11 Nov. 1986: 1.
4 Martin, Sue. "TV Review: 'Star Trek' Pilot Airs on KCOP." Los Angeles Times. 5 Oct. 1988: 8.