Elizabeth and Lady Catherine actually have a couple of conversations. There is especially a long, witty exchange when they first meet. However, since you do not specify, I am assuming you are actually referring to the dramatic conversation towards the end of the novel, in which Lady Catherine visits Longbourn with the soul intention of commanding Elizabeth not to marry Darcy (Ch. 14, Vol. 3).
Lady Catherine begins this conversation with the condescending remark that the Longbourn estate is small and then further insulting remarks about their sitting room. When she and Elizabeth are alone together she states that the reason she traveled to Longbourn was that she had heard very startling news. First, she heard that Jane was engaged to be married to Bingley. Secondly, she heard that Elizabeth would soon marry Darcy. Lady Catherine then makes the ironic statement that while she knows the news is a lie, she traveled to Longbourn to find out from Elizabeth that the news was indeed a lie.
Always the wit, Elizabeth responds by saying that the very fact that Lady Catherine traveled so far is actually proof that the news is true, if there really is such news.
Lady Catherine then demands Elizabeth to assert "that there is no foundation" for the news that Elizabeth is to marry Darcy. Elizabeth states that she will not answer the question. During some further repartee Lady Catherine explains Darcy's betrothal to her daugher and states that Elizabeth is socially beneath Darcy. Finally, Lady Catherine commands Elizabeth to promise that she will never "enter into such an engagement" with Darcy and Elizabeth flat out refuses.