Bribing is a mechanic in which antagonistic students force Jimmy to pay them money so they won't beat him up. The bribe is $1. Bullies, Preppies, Greasers, Jocks and Townies may demand bribes, Nerds and Non-Clique Students don't. 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 non-Nerd clique member will demand money. The introduction mission Welcome to Bullworth starts with Trent and Ethan demanding bribes from Jimmy.
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 passes English 5, his insults are good enough that students will usually back off if he insults them. The largest ones may still demand a bribe depending on Jimmy's actions and his faction standing with their clique.
If Jimmy antagonizes another student by messing with him or his clique, that student may become angry and demand a bribe. It is unclear what causes students to demand bribes instead of just attacking when they become angry.
Earlier in development, Jimmy would be able to bribe the Prefects to leave him alone.