What does "rumble" mean to each of the Greasers? How is having a "rumble" going to solve the problem between the Greasers and the Socs?

A rumble is simply a colloquial term for a large-scale gang fight. For the Greasers, no less than the Socs, it's a matter of pride. A rumble is a way of settling scores, of proving which is the stronger gang. To the Greasers, a rumble isn't simply an unseemly street brawl; it's more like an old-fashioned duel, a matter of honor between two combatants carried out according to certain rules. The Greasers may live a rough life, but when it comes to fighting, they prefer what they consider a fair fight to a rough one. As Two-Bit explains to Cherry and Marcia:

''A fair fight isn't rough,'' Two-Bit said. ''Blades are rough. So are chains and heaters and pool sticks and rumbles. Skin fighting isn't rough. It blows off steam better than anything.''

The forthcoming rumble, scheduled to take place in a vacant parking lot, is intended to be a "fair fight." In other words, no one will use weapons, only fists. Even so, one of the Socs, Randy, still doesn't see what the rumble will accomplish. He's come to realize that violence only begets violence: it doesn't really solve anything. All the same, he's worried that the other gang members will think of him as a wimp if he doesn't get involved.