Instructions
Obtain a Water Stone if you want to evolve Poliwhirl into Poliwrath. If you're playing "Pokemon Black" or "Pokemon White," you can get a Water Stone in Castelia City. Talk to the scientist there and answer "Panpour" when he asks you a question. He gives you a Water Stone.
Select "Water Stone" from the inventory and give it to Poliwhirl to hold. As soon as you do this, Poliwhirl turns into Poliwrath.
Get a King's Rock if you want to evolve Poliwhirl into Politoed. In "Pokemon White" and "Pokemon Black," find a King's Rock on Route 13. Some wild Poliwhirls are already holding a King's Rock when you catch them, but this is somewhat rare.
Go to a Pokemon Center to trade the Poliwhirl while it holds a King's Rock to a friend who also has a "Pokemon" game. During the trade, Poliwhirl turns into Politoed. Have your friend trade Politoed back to you when the evolution completes.