Rock, paper, scissors is much more important in Japan than in the west. Since decision-making takes forever in Japanese Schools, it's usually easier to just play this one time to decide who goes first.
How To Play, the Japanese Way
First, you say Saishou Gu, which means "first, rock" while moving your fist down on Sai, up on shou, then down on gu again. Then you pull back and strike the air with the fist again while saying Jan. Then pull back for your big windup on ken- and decide which of rock, paper, or scissors you will make. Do it as you come down on the word pon.
if there's a tie, then say "aiko deshou" and do it again until someone loses.
Japanese kids are great as this - you know how in America, every little kid can do a high five? That's how jan-ken is in Japan, starting at age 2 or so. There are some other variants - the fast version, the version where you don't move your hands. There are also many kids who cheat at jan-ken-pon, by waiting until they see what you pick before they pick themselves.
Not Just Two People Anymore
There are some other uses found only in Japan - while at home you usually play with only two people, in japan you can play with any number of people. So with 5 people, if there are losers they sit out, and the rest do it again. If you're trying to find a winner, it's easy, but if you are trying to find a loser to do some chore, it can be confusing - the initial losers sit out, but the person who loses last is usually considered to be the real loser. I've even seen it played with 8 or more people. It takes a long time.
You can also use it for deciding teams - someone will say "only rock or paper" and then everyone does it until half are one and half are the other. There's no being picked last in gym class phenomenon in Japan, since they always use this to decide scrimmage teams.
A Strange Phenomenon
Sometimes, there are about 10 ties in a row with just two people. There is a certain strategy of doing rock, then paper, then scissors, and if both people employ it, then they will tie for a long time. It's played really fast though, so while this is going on if you want to break out of it, you have to think fast and think what the other guy's going to do next, and then think what beats it.
Finally, jan-ken-pon is used when there is a tie in some other game- so if two people try to grab something at the same time and start fighting, one will start the sai-shou chant and the other will be forced to agree to arbitration.