Elizabeth's quickness of mind, or intelligence, is expressed through two modes: wit and discernment. The reason why Elizabeth's intelligence is essential to the novel is that Austen uses the novel as a social criticism, particularly critiquing character traits. Elizabeth stands in great contrast to her other sisters and other characters deemed in the novel as ridiculous. Elizabeth particularly stands in contrast with Jane who lacks an ability to judge and discern, but rather sees everyone and every situation as benevolent. Austen uses Elizabeth's mind to say things in general about judgement--sometimes it is necessary to judge a character as untrustworthy, such as Wickham, and sometimes missing that important judgement is disastrous. In addition, sometimes mistakenly judging someone, such as Mr. Darcy, can also have tragic consequences.
Elizabeth's quickness of mind is particularly seen in her witty comebacks. Many can be seen all throughout the novel. Some in particular can be seen while she is staying at Netherfield while tending to her sister. For instance, Elizabeth gives a very witty retort after Darcy proclaims that a woman can only seriously be considered "accomplished" if she has:
a thorough knowledge of music, singing, drawing, dancing, and the modern languages ... [plus] possess a certain something in her air and manner of walking, the tone of her voice, her address and expressions. (Ch. 8)
Elizabeth's retort is, "I am no longer surprised at your knowing only six accomplished women. I rather wonder now at your knowing any" (Ch. 8). A second instance of witticism can be seen when, while Miss Bingley is playing the piano, Darcy asks Elizabeth if she would like to "dance a reel," meaning dance a jig, and Elizabeth replies, "You wanted me, I know, to say "Yes," that you might have the pleasure of despising my taste; but I always delight in overthrowing those kind of schemes" (Ch. 10).
As stated above, other than witty comebacks, we also see Elizabeth's quickness of mind in what she believes is her ability to discern. She believes she has the ability to correctly judge character, as we see when she judges the Bingley sisters as unworthy of knowing, judges Darcy to be prideful and arrogant, and judges Wickham to be the best man she has ever met. However, we soon learn that she is wrong on two accounts. Wickham turns out to be a scoundrel, and Darcy turns out to be the best man of her acquaintance.