It's not what Dream World does to your Pokemon's ability, but the Pokemon you meet that have special abilities only obtainable through the Dream World!
Registering your Black and White game into the website allows you to go to the Dream World online, where you play games, get items, and meet Pokemon with special abilities that are only obtainable through it!
After playing a mini-game with them, you are able to "wake up" your Pokemon sent from your game via Wi-fi, and you will receive anything you have obtained playing in Dream World such as berries, and the Pokemon you encountered! You may encounter more than one, but you will only be able to choose one.
You can do this as many times as you wish (as long as the site isn't under maintenance, every Tuesday, and various other maintenances)