Short for Reputation.
The T-Birds always wanted to protect their `reps', and would threaten those who might intentionally or inadvertently damage said `rep'.
The T-Birds reputations were as tough, `Don't Take No Crap From Nobody' bikers. In actuality "They were just kids pretending to be tough when they actually weren't", as per Adrian Zmed on Facebook.
Louis: (Softly, to Michael) Listen, when we make the drop, not out in the open like this, huh? I got a rep to protect.
Goose: Remember, this is between us. I got a rep to protect, okay?
Johnny: Remember. I got a rep to protect, OK, Shakespeare?
Michael and Davey in a deleted scene Davey: but we gotta keep this strictly between us, ‘cause…
Michael: `You got a rep to protect’.
Davey: Uhh, well… yeah.
Johnny: (To Stephanie) I got one more thing to say to you. That jacket you are wearing is T-Bird property. You wanna leave the party? Drop the jacket at the door on the way out. I got a rep to protect. So don't damage the rep, and we don't damage new friends, dig?