To be strait-laced is to be extremely strict and uptight. Your strait-laced cousin won't be interested in joining the rest of your family in an impromptu dance party.
If you know anyone who's extremely prim and proper, you can describe them as strait-laced. This adjective originally described overly tight clothing, particularly women's stays or bodices. The now obsolete adjective strait meant "narrow or cramped" or "strict," but because it's become unfamiliar today, strait-laced can also be spelled straight-laced.