Madame (Socialite until Gen VI) is a Trainer Class introduced in Generation IV. They are old women wearing expensive outfits. They use Pokémon based on pets. Their male counterpart is a Gentleman. They are the much older versions of Ladies, they give out huge amounts of money when defeated. In Generation VI they appear in the Battle Chateau under the names of Baroness, Viscountess, Countess, Marchioness and Duchess. They are only called "Madame" when battled in the Battle Institute or Battle Maison.
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There is a Socialite by the name of Reina who is the second richest Trainer in Pokémon Diamond, Pearl and Platinum. Along with Gentleman Jeremy, they are the second richest trainers in the game (with Amulet Coin).