Bribing is a game mechanic in which antagonistic students force Jimmy to pay them money so they won't beat him up. The bribe is $1. If Jimmy tries to walk away from students demanding a bribe they'll follow him, and if he stands around without paying, they'll attack him instead.
How likely a student is to demand a bribe depends on how big they are, how antagonized they are, how many English classes Jimmy has passed, and his clique's respect toward Jimmy.
At the beginning of the game, any male clique member excluding the Nerds will demand money. The introductory mission Welcome to Bullworth starts with Trent and Ethan demanding bribes from Jimmy. In the mission This Is Your School, the game forces Jimmy to pay Russell $1.
After Jimmy passes English 1, he gains the ability to apologize to most antagonistic students without being forced to pay, although larger ones like Russell, Bif, and Damon will continue demanding bribes.
Once he completes English 5, his insults are good enough that students will usually back off if he insults them. Larger ones like Russell, Bif, and Damon will only demand bribes if his respect toward their faction is below 50%.