These clauses are known as the standard clauses. They take place in every battle, and promote fairness and skill.
Uber Clause: You are not allowed to use more than one Pokémon considered Uber on your team. There will be a tier list set up soon; for now, just head to
Smogon's tier list.
Species Clause: You may not have more than one of the same Pokémon on your team. I believe this includes evolutions (for instance, no Magneton and Magnezone on the same team).
Sleep Clause: No two Pokémon can be asleep in a team at the same time, with the exception of self-induced sleep.
Item Clause: You are not allowed to use certain items deemed "hax" (including King's Rock, Red Card, and BrightPowder). The Pokémon in your team are not allowed to hold more than one of the same item, with the exception of Leftovers and Life Orb.
Evasion Clause: Moves that increase evasiveness or lower your opponent's accuracy are banned. This ends up making the battle rely solely on luck, which is not a good indicator of skill.
OHKO Clause: One-Hit Knock Out moves are not allowed. Specifically, you are not allowed to use the moves Fissure, Sheer Cold, Guillotine, or Horn Drill. The 30% accuracy and instant-kill that results also ends up making the battle rely on luck.
Self-KO Clause: In the rare event that both you and your opponent are on your last Pokémon, and one competitor uses a move like Explosion to attempt to induce a draw, the person that initiated the move is considered the loser.
Hack Clause: You are not allowed to use hacked Pokémon. Gym Leaders hold the right to checking said Pokémon after a battle.
Along with the above standard clauses, the Gym Leaders also have an extra clause:
Super-Effective Clause: Upon entering a Gym battle, you can not have more than two Pokemon that are directly super-effective to the Gym Leader type.