Unlike Pokémon battles, which take place throughout the Hoenn region, Pokémon Contests involve going to specific contest halls and going through two rounds of judging based on the condition of Pokémon and the moves they used in a performance exhibition.
To enter your first contest, you have to obtain the Contest Pass from one of the people in the Normal Rank Contest Hall in Verdanturf Town, although the first contest hall you will find in a play-through is the Hyper Rank in Slateport City. You obtain the PokéBlock Case, which lets you give your Pokémon condition enhancing PokéBlocks. PokéBlocks are made with berries and through playing a timing based single player or multiplayer minigame.
Pokemon Contests have Pokemon each use one move to gain the appeal of judges, or to disrupt other contestants. They have 4 chances to make appeals.
Enhancing the condition of a Pokémon with PokéBlocks is critical for higher level competitions.
You might want to pick a Pokémon that has moves that are the same type as the contest they are participating in.
If you are having trouble winning contests or want to get very competitive, you may even consider raising a Pokémon specifically for contests instead of one for battle.