The 5 standards are Uber, OU, UU, RU and NU (possibly 6, is Little Cup standard?) And you can play stuff up but not down. Smogon's site has a list of their standard tiers.
The clauses you'd probably see:
Sleep: Only 1 Pokemon per team can be asleep
Evasion: Evasion boosting moves/abilities are banned, except (I believe) Acupressure. Never actually used Acupressure myself, I find it's a pretty poor move...
Moody: Moody is banned.
OHKO: OHKO moves always miss
Selfdestruct/Explosion: If one of these moves KOs both people's last mon, the Exploder loses. Applies to Destiny Bond too.
Drizzle and Swift Swim can't be on the same OU team either, but you can get around that with Rain Dance instead of Drizzle (tougher to pull off but rewarding if you do).
Did I miss anything there?
Oh, and my own question: PO tiers are slightly different from Smogon standard, so Showdown and PO have slightly different OUs... If on PO, do we still follow Smogon, or do we follow PO? (Don't much care either way, I hate Genesect in OU and don't use it even on PO outside Ubers, just curious)