What happens between Mr. Collins and Charlotte Lucas?

Mr. William Collins has a grand scheme to help himself and to help his cousin, Mr. Bennet. A clergyman, Mr. Collins has been encouraged by his patroness, Lady de Bourgh, to set a proper example for the parish by marrying. As the holder of the entail to the Bennet estate—as the one who will legally inherit the estate of Longbourn because Mr. Bennet has no male heir, only daughters—Mr. Collins hopes to provide for the Bennet women after Mr. Bennet's death by marrying one of girls. This way he will honor his patroness's wishes and simultaneously save his cousins from becoming homeless.
Told that Jane is on the verge of an engagement, Collins sets his sights on the second eldest, Elizabeth. Collins has so much pride because of his perceived success in the world and because of the magnanimity of his offer that he cannot imagine any of his cousins turning his offer of marriage down. He is only mildly deterred when Elizabeth refuses him.
The next day [after the Netherfield ball] opened a new scene at Longbourn. Mr. Collins made his declaration [of marriage] in form. Having resolved to do it without loss of time, as his leave of absence extended only to the following Saturday .... "I am persuaded that when sanctioned by the express authority of both your excellent parents, my proposals will not fail of being acceptable."
Collins is brought to a reconsideration of his proposal by Mrs. Bennet's remarks: "Lizzy shall be brought to reason. ... She is a very headstrong, foolish girl." He says, "[If] [Elizabeth] is really headstrong and foolish, I know not whether she would altogether be a very desirable wife to a man in my situation."
While Mr. and Mrs. Bennet continue to work out the situation with Elizabeth ("Your mother will never see you again if you do not marry Mr. Collins, and I will never see you again if you do"), Charlotte Lucas, ventures into the confusion. Charlotte makes herself useful for several days by paying attention to and listening to Collins. Yet Charlotte's real aim was to "secure ... Mr. Collins's addresses ... towards herself."
Charlotte's motives for engaging Mr. Collins' affections is one of the controversial elements of the novel. Some see it as strictly a marriage of convenience; some see it as an act of greed. Elizabeth sees it as the sacrifice of "every better feeling" within Charlotte. What does Charlotte see it as?
Charlotte is the eldest daughter of a man who holds a knighthood but has very little wealth. She is not beautiful like Jane nor as witty as Elizabeth. She is not able to go to London during the social season to meet men who might choose to marry her. Her prospects are limited to Meryton and the village of Longbourn where, at best, she might meet officers who find her too serious.
Charlotte believed that if she were ever to have any independence from her parents and a home and family of her own, her best chance lay with Mr. Collins. True, she saw that he was silly and vainly proud, yet, she saw too that he had a good position in his profession, a good yearly income, a good social standing with his patroness, and a prestigious place in Lady de Bourgh's village of Rosings. Charlotte traded obscure dependency with her parents for independence and her own comfortable home and family with Mr. Collins. She was content. He was happy. It turns out that Charlotte was exactly suited to adapt to a life in which she could make advantages outweigh disadvantages.
"I am not romantic, you know; I never was. I ask only a comfortable home; and considering Mr. Collins's character, connection, and situation in life, I am convinced that my chance of happiness with him is as fair as most people can boast on entering the marriage state."