When I learned this game in Cape Verde, the kids just called it Salta ('Jump'), and it was the game all the little girls played. All the English speakers called it African jump rope (or more often 'that jumping game with the string'). I didn't meet anyone who called it Chinese jump rope, but I've come across it a couple times in America since then, always under this name; in Europe, it is given names like elastics, French skipping, and Gummitwist.
The Basic Idea: You will need three players and a loop of string, rubber bands, or elastic rope. Two players stand inside either end of the loop with their ankles spread slightly apart, the loop pulled tight so as to raise it 3-4 inches off the ground. The third person hops over this rope in a predetermined pattern. If they hop the pattern correctly, the loop is raised up to knee level, and they try the pattern again. Next is hip level (AKA hipsies or underbums); obviously you are not hopping at this point, but leaping. (I have never played with anyone whose hip level was higher than mid-thigh on me, but even so I was exhausted after one round).
Details: The rules, patterns, and penalties vary from player to player, game to game, and often minute to minute. Here are some of the more common variables.
The most common foot placements from which to make patterns are: feet-out-same-side, feet-out-on-opposite-sides, both-feet-in, one-foot-in-one-out (and vise versa), both-feet-out-opposite-sides-but-with-legs-crossed. To these might be added standing on the loop (this doesn't work very well above ankle level), catching part of the loop with your toe, and one-foot hops and jumps.
The loop of string may be crossed in the center (making a figure eight), allowing for more complex patterns. It may be held higher at one end than the other. Turns and spins may also be added. Footwork can get pretty fancy, especially among older kids.
The pattern is usually accompanied by chants, rhymes, and/or clapping. The chants often describe part of the pattern to be hopped. ("England, Ireland, Scotland, Whales; inside, outside, puppy dogs' tails!"1). On the other hand, in my village we just used the days of the week. Oddly, there doesn't seem to be much overlap between Chinese jump rope rhymes and everyday jump rope rhymes.
Whether or not you are 'out' depends (usually) on the judgement of the two people playing 'post'. You may be called out for touching the rope, or you may be able to do anything short of falling flat on your face without being called out. Usually it falls somewhere in between, with touches allowed, but trips, catches and stumbles ending your turn. Sometimes it will not be kicking the rope, but breaking rhythm that will be the cause of outs. Often you start your second turn at whatever point you messed up in your first turn.
You can add in extra height levels; ankle, calf ('shinnies'), thigh, underbum, and hip. If you're particularly athletic, you can also do under-arms and neck, although at that height one usually does high-kicks rather than jumps. Wikipedia claims that 'most people' use their hands to perform the patterns once the rope moves too high, kneeling under the loop. This seems to be a comparatively uncommon variation, as far as I can find.
Winning: Obviously, you could play this game in such a way that someone wins, and everyone else loses. You could count the number of outs, or outs could be final, with no second chance. I'm sure that some people play it that way. But I've never seen it played like that, and from what I can find on-line, Chinese jump rope doesn't usually have winners and loosers. The game keeps going until everyone is either exhausted or bored. Being out only means that your turn is over; you'll get to try again after the others have gone out. It is certainly a competitive game, in that everyone wants to be the best at it, and you can easily see who is doing better than you are, but it's not usually a winning type of game.
tWD says: Spastic elastics, that's what we called it.
1 Rhyme from http://folksong.org.nz/pdf_copies/Bauer_elastics.pdf
Ana, Otru Ana, Edi, Katia, Sara, Ne, Aneti, Jileti e kel otrus di Hortelao.