What is a ninth-born son?
Exactly what it says on the tin the ninth born son of someone. They are most likely from some noble House instead of low born but the quote doesn't really give anything away either way.
Is it a crime?
No, only the first born son in a House in Westeros will inherit and become Lord, the others must find something else to do. This related question discusses it in more detail but in short the options are.
- Night's Watch
So out of those options why would someone choose the Night's Watch? @Aegon has a good answer for that here so I won't repeat too much. Another reason though is it was once seen as an honourable choice to serve in the Watch, it's only really in recent years that's changed though some people like Jon Snow and some other high born sons still think it's an honourable thing to do.