LUKE: Well, ’cause apparently he’s been getting into some trouble and Liz is afraid he’s heading for something bad, and rather than handle it herself, she’s just giving up. She’s sending him here so I can straighten him out.
LORELAI: Right. So what kind of trouble has he gotten into?
LUKE: Ah, just kid stuff, you know, staying out late, getting rowdy. I don’t know exactly.
The exact nature of the “trouble” Luke’s teenaged nephew was getting into was never made explicit in the show, so that the characters were free to imagine it for themselves.
Luke optimistically guesses that his nephew has been getting up to trouble in an almost boyishly wholesome way that could have come straight from the pages of Tom Sawyer. Lorelai immediately assumes that the kid is fresh out of juvenile detention and most likely has a criminal record.
Whether either are right, or whether the nephew’s misdemeanours are somewhere between these marks, is something the viewer is free to decide for themselves.