In White you get Triple battles. Black gets Rotation battles. The first time you can use these battle modes is in Driftveil City, against a Biker riding back and forth next to the market. He will explain the rules to you, then ask if you want to have one with him. There is at least 1 more with a certain Trainer on Route 7 I believe it was.
As for link play, you may be able to do both from both versions, I'm not sure. I haven't used it at all so I don't know.