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There are several different theories as to the origin of the phrase, but there is no definite proof as to which is correct.
One explanation suggests that "Ps and Qs" is short for "pleases" and "thank-yous", the latter of which contains a sound similar to the pronunciation of the name of the letter "Q". This phrase would be used by parents to educate their children to not forget to use those polite words when they speak to people. Possibly, it meant "and "excuse me." Young children would pronounce them as Ps and Qs.
Another origin comes from English pubs and taverns of the seventeenth century. Bartenders would keep a watch on the alcohol consumption of the patrons; keeping an eye on the pints and quarts that were consumed. As a reminder to the patrons, the bartender would recommend they "mind their Ps and Qs". This may also have been a reminder to bartenders not to confuse the two units, written as "p" and "q" on the tally slate.
Other origin stories, some considered "fanciful", could come from French instructions to mind one's pieds (feet) and queues (wigs) while dancing. However, there is no French translation for this expression. Another origin could be from sailors in the eighteenth century who were reminded to pay attention to their peas (pea coat) and queues (pony tail).
Another possible and viable theory is after the Norman Invasion of 1066 the courts, church, and establishment were becoming French speaking and the English dialect of the 11th Century had no qs; so one must watch their usage in court or discourse with the French Norman conquers. 
A possible origin or at least similar expression comes from seventeenth-century slang. "P and Q" meant "prime quality" or "highest quality". It seems unlikely that the phrase "P and Q" stood for "prime quality", because that does not explain the presence of the word and.
It is also possible that the expression refers to the careful reading of Medieval Latin texts: the letters "p" and "q" had various scribal abbreviation symbols for different shortened words. For example, "q" with a dot over it was the abbreviation for quod while "p" with a line through the tail of the letter was the symbol for per. Minding that these abbreviations were interpreted accurately (i.e. that one read "per" as opposed to "post" or "pro") would ensure the correct reading of the text.
Another origin of the story of "mind your Ps and Qs" comes from early printing presses. Printers placed individual letters on a frame to print a page of text. The letters were reversed, making it easy to mistake lowercase ps and qs in setting the type and were fined for every spelling mistake, hence the personal financial importance of accuracy. A reminder to stay watchful of the details could have come from this time as well. In a similar setting, this expression has been attributed as an adage for teaching children to spell.
David Covey's book on improving quality and success in a consumer-driven world is titled Mind Your Ps & Qs, using key phrases starting with "p" and "q" to illustrate his points. Edward Makhene has written a philosophy book titled Mind Your Ps and Qs: Philosophy Goes to High School with the goal of having students study and analyze society's common-held beliefs and their validity. Brief origin stories can be found in linguistics books, like Karlen Evins' I Didn't Know That or David Wilton's Word Myths. The expression can be found used by journalists in headlines as well, often based upon the "manners" definition, as can be seen on MSNBC.