A new minigame introduced in FireRed and LeafGreen is Berry Crushing. While similar in design to Berry Blending in Hoenn, Berry Crushing serves a different purpose.
In order to take part in the Berry Crushing, you first must collect a Powder Jar from a man in Cerulean City (just north west of the Pokemon Center). From there you can connect with 1 to 4 other players in a Union Room to grind up berries. This even includes Emerald players, who instead collect the Powder Jar from a woman in Slateport City's market.
Using Berries collected on the field and from wild Pokemon is how it's done. The key to getting the most from your Berry Crushing is to tap the A-button as quick as possible. Having more players involved will exponentially increase the amount of powder you receive. Finally, different berry combinations seem to yield higher amounts of powder.