It's pretty well accepted that something crashed in the New Mexico desert July 7, 1947 - only a few days, really, after pilot Kenneth Arnold said he saw a flying saucers near Mt. Ranier on June 24, which was the first widely reported sighting. Then, on July 8, the Roswell Daily Record reported the press release by the Army stating that a flying saucer had crashed in the area. The next day, the military changed the story to suggest that it was a crashed weather balloon, and not an alien craft. The story died down until, a little more than 30 years later, physicist Stanton T. Friedman interviewed Major Jesse Marcel, who was involved in the original debris cleanup and claimed the crash was no balloon.
Although it wasn't the beginning of the UFO fascination in the States (that probably has a lot to do with Arnold, and Edward R. Murrow's CBS program, "The Case for the Flying Saucers"). But Roswell is the UFO incident most people withing the mainstream know about, and it did solidify the government cover-up conspiracy theories.
To learn more about the Roswell story, check out Lee Speigel's excellent two-part story on the crash and the festival at AOL News.