e.g. Yache Berry, weakens a foes supereffective Ice-type move.
You can only get berries in dream world. Once you enter dream world you can plant berries and they will grow into more berries the you can send back to your game via entree forest. You can get berries by checking rustling bushes in pleasant forest.
You can find them in the dream world as everyone has said, but there are also a few pokemon who carry them rarely, such as raticate
you can get most berries in dream world, but for in-game solutions, you can get some as prizes for landing on bonus areas in spin trades.