Mistress Quickly is the hostess of the Boar's Head Tavern in Eastcheap, London, which is the kind of joint frequented by criminals, prostitutes, and assorted disreputable figures (like Prince Hal and Falstaff). In other words, she won't be sipping tea as a guest at the royal palace any time soon.
We don't see much of Mistress Quickly in Henry IV Part 1 (her character is developed further in the plays Henry IV Part 2, Henry V, and The Merry Wives of Windsor), but she does participate in the famous skit that mocks King Henry IV in Act 2, Scene 4 and gets into an argument with Falstaff over whether or not she picked his pockets while he was passed out in the corner of the bar. (She didn't, but stage actors often portray her as the kind of woman who is not above riffling through the pockets of unconscious bar patrons.)
Stage and film productions tend to play up Quickly's status as a "sits on the lap of strange men" kind of girl (check out this photo from one production of the play). Her bawdy name recalls the kind of brief sexual encounter one might experience in an Elizabethan brothel or tavern. In Elizabethan England, taverns and brothels were sometimes synonymous and Shakespeare is famous for giving characters like Mistress Quickly bawdy names (like "Mistress Overdone" and "Doll Tearsheet"). Although the hostess is married in this play (Shakespeare later kills off her husband so she can party with the other singles), her name hints that she's been around the block, a few times.
Because of her association with the raucous and topsy-turvy world of the tavern, Mistress Quickly is an important figure that embodies the play's rebellious spirit. In fact, all three female characters in the play are associated with some form of rebellion. (Check out our discussion of "Gender" and "Rules and Order" for more on this.)