What does it mean to mind your P's and Q's?

Solution 1:

"Mind your Ps and Qs" means "be careful to behave well and avoid giving offense."
The NOAD reports that its origin is unknown; it would refer to the care a young student must take in differentiating the tailed letters p and q.

Solution 2:

I've always heard that it was originally short for "Mind your Pints and Quarts!" Something an innkeeper (or what-have-you) would shout to an unruly common room to settle them down. According to AUE no one's particularly sure, but they list a few other theories, including what they consider to be the most likely: "Mind your 'please's and 'thank you's".

Link: http://alt-usage-english.org/excerpts/fxmindyo.html