In Japan (as in most of East Asia) a crane represents longevity, because it was believed to live for a thousand years. Folding a thousand of them, one for each year, is supposed to grant one a wish for good health, or when given as a wedding present, a long and prosperous marriage. Or, thanks to the story of Sadako and the Thousand Folk Processes, a prayer for world peace. From Japanese 千葉鶴, being sen, thousand + ha, the counter for pieces of paper + tsuru, crane, with sound-changes from the concatenation, because that's the way Japanese rolls.
Instructions for folding a senbazuru available at http://www.monkey.org/~aidan/origami/crane/, and many other places around the internet.