Charlotte Lucas spends time talking to Mr. Collins. Elizabeth thinks Charlotte is simply being a good wingwoman, never dreaming that her friend is hoping for a marriage proposal.
Mr. Collins proposes to Charlotte on the Saturday morning before he leaves for Hunsford, and she accepts.
Her acceptance is on purely practical grounds. He's a good man, with a good career, and she knows she is unlikely to receive other, better offers. She is simply worried about how Elizabeth will respond.
The whole Lucas family is overjoyed.
Mr. Collins informs the Bennet family that night that he will accept their invitation for a speedy return visit, surprising all of them. They think he means that he has turned his attentions towards one of the younger Bennet girls and they know that Mary would be willing to accept. (She is plain and bookish and unlikely to get any other marriage proposals.)
The next morning, though, Charlotte arrives and confides in Elizabeth that she is engaged to Mr. Collins. Elizabeth is surprised but she wishes Charlotte happiness.
Charlotte explains that it is a practical decision. She is not romantic, but she does want a comfortable home. Mr. Collins can provide that.
When Charlotte goes away, Elizabeth reflects on the conversation. She had always known that Charlotte's understanding of marriage was different than her own, but she finds it hard to believe that Charlotte would throw away every possibility of emotional satisfaction just to be married.