Someone who's straitlaced is very prim and proper. Your straitlaced parents will be alarmed if you dye your hair purple and drop out of school to become a mime.
The adjective straitlaced, which can also be spelled strait-laced or straightlaced, is a synonym for "uptight" or "stuffy." The general idea is that straitlaced people are too rigid or narrow in their thinking, based on the old-fashioned adjective strait, "tight," and the fashion of lacing women's dresses extremely tightly. Phrases like "strait and narrow" and the word "straitjacket" share this "tight" root with straitlaced.