Official Hallowed Competitive Fantasy Battling Tiers
Uber Tier
OU Tier
UU Tier (anything not listed in Uber or OU)

Clauses
Species Clause - A player cannot have more than one Pokemon with the same National Pokédex number on a team. For example, using two Beedrill on the same team is not allowed.
Evasion Clause - A Pokemon may not have the moves Double Team or Minimize in its moveset.
Sleep Clause - You lose if, at the end of your turn, two of your opponent's Pokemon were put to sleep by a move you used but were not forced into, and neither have had their sleep removed since then. Rest does not count towards this rule.
OHKO Clause - A Pokemon may not have the moves Fissure, Guillotine, Horn Drill, or Sheer Cold in its moveset.
Swagger Clause - A Pokemon may not have the move Swagger in its moveset.
Shadow Tag Clause - A team cannot have a Pokemon with the ability Shadow Tag.

Rules
Tiers - You may only use Pokemon from the same tier and tiers below the Tournament/Gym's listed tier. For example, if a Tournament is an OU Tournament you cannot use Uber Pokemon but can use Pokemon from all tiers below it.
Levels - You may not use Pokemon over the Tournament/Gym's listed level cap and all of your Pokemon must have leveling disabled, unless stated otherwise.
Items - You are not allowed to use any restoration items during battle, such as Potions and Revives. You may use held items such as Life Orbs, however.
Disconnecting - Disconnecting mid-match during an official battle will result in a forfeit, unless ruled otherwise by a Tournament Official or Gym Leader. Disconnecting mid match will result in a forfeit.
Pre-Battle - Your Pokemon are not allowed to have a status before the start of your battle. For example, they can't be poisoned, burned or have any other status before you begin your battle.