The skill of ball-juggling - to keep a ball in the air for as long as possible by bouncing it off any part of the body that is allowed in the rules of football, that is, any part except the hands and arms. Typically the ball is kept up using the feet and head.
Many of the phrases that are explained on this site have origins definitively locating them in the United Kingdom. In keeping with England being, obviously, the source of English, in most cases those phrases are old. In this case though the phrase is quite recent and the source is the north of the UK - Scotland.
The game is similar to other ball games but the version specifically known as keepy-uppy (or keepy-uppie) began in Scotland in the post WWII period. The pastime is one that footballers have long used to develop their skill and football's greatest player, Pelé, was especially adept at the game - called then Freestyle Football. It becomes harder to keep the ball in the air using a smaller ball. The Argentinian star Diego Maradona was renowned for ball juggling with a golf ball. He also used to entertain football crowds at half-time with the Maradona 7, a juggling skill using just the right foot, then left foot, then right and left thighs, then shoulders, and finally just the head.
Before the name migrated to keepy-uppy it was called keepie-up and was referred to as that in the Glasgow newspaper The Sunday Post, in February 1958:
"He's just jealous because I aye beat him at keepie-up!"
The first known reference to the game as keepy-uppy comes from The Times, November 1983, although it had certainly been called that in school playgrounds some years earlier:
"I used to play keepy-uppy, all that sort of thing. You become the master of the ball."
The pastime has grown from a simple skills practise in which children try to outdo each other with the number of keepy-ups they can manage (almost always using just the feet for novice players) into feats that get into the Guinness Book of World Records. These have now gone well beyond a few hundred kicks and records are now measured in hours rather than numbers of bounces. At the time of writing the record stands at 19 hours 30 minutes.