The conflict which is probably most significant to the plot is the one between Elizabeth Bennet and Fitzwilliam Darcy. She finds him to be proud and haughty; at first, he doesn't find her terribly attractive, and—even when that changes—he cannot escape his feeling that she is inferior to him, at least socially. Darcy is conflicted within himself even, which he reveals during his first proposal of marriage to Elizabeth. Darcy doesn't want to love Elizabeth, because she and her relatives are so far beneath him in his opinion, and her mother and youngest sisters are especially inappropriate. He gets over it when he realizes that he has not behaved in a gentlemanlike manner.
The conflict between Mr. and Mrs. Bennet certainly informs their two eldest daughters' ideas about marriage. Mr. Bennet married Mrs. Bennet when they were young, before he really got to know her, and she is absolutely ridiculous and obnoxious. She believes that he has "no compassion" for her nerves, constantly making her anxious, and he does his best to avoid her company whenever he can, though he does seem to enjoy making her suffer a little. Elizabeth and Jane, consequently, really want to marry for true love.
There is also the conflict between Elizabeth and Miss Bingley, who is clearly trying to capture Darcy for herself, though she goes about it all wrong. Miss Bingley is a total snob, who doesn't want her brother marrying Jane Bennet either. Elizabeth also conflicts with Lady Catherine, Darcy's aunt, when Lady Catherine hears that Darcy has proposed to Elizabeth. Lady Catherine feels that the Bennets are well beneath her family, and she has her heart set on Darcy marrying her own daughter, and his first cousin, Lady Anne.