"The primary function of the Académie will be to work, with all possible care and diligence, to give specific rules for our language and make it pure, eloquent, and useful in the arts and science."
Article 24 of the Bylaws of L'Académie française

Mission and Role
The Académie's primary mission, maintenance of the French language, was the only one specified in its original bylaws. To achieve this goal, the Académie has historically worked to fix the language, giving a linguistic heritage to all French citizens and everyone who speaks the language. Today, it works to maintain high standards of usage while keeping up with the necessary evolution of language. These standards are maintained by the production of the official dictionary, and by recommending updates and participating in the creation of new terms. The secondary mission of the Académie, literary patronage, was not part of its original responsibility. It has only been made possible by gifts and grants, which enable the annual distribution of twenty-five literary prizes. Of particular note is the Francophone Grand Prize, given annually since 1986, which recognizes those who have helped the Académie's primary interests by spreading the French language globally. Subsidies are also given to literary companies, families in need, and students.

History
The Académie was founded in 1635 by Cardinal Richelieu. The bylaws he drafted were signed by Louis XIII that same year, and registered with Parliament in 1637. Richelieu's intended mission for the Académie was that it would fix the language and give it rules, making it pure and able to be understood by everyone - at the time, it was a vulgar language, with dozens of vernaculars spoken throughout the country. To achieve Richelieu's goal, the Académie published its first dictionary in 1694 (the latest is the ninth edition of 1992). Richelieu was named the Académie's protector, and after his death the job passed to Chancellor Séguier. Following that, Louis XIV became the next protector of the Académie, and since then the job has belonged to the successive kings, emperors, and heads of state of the country. Over the centuries, members have met at a variety of locations, including the homes of individual members. Offices were first established in 1639 for the publication of the first dictionary, but these too were in the homes of Chancellor Séguier, Perrot d'Ablancourt, and Sirmond. After Séguier's death, the Académie met in the Louvre until 1806, when it moved to the old building of Le collège des Quatre-Nations.

Members
There have always been 40 members of the Académie, and a total of 700 over its existence. They are poets, philosophers, doctors, linguists, art critics, heads of state, clergymen, and from other fields - the varied composition being designed to provide a wide range of knowledge and culture. They are charged with judging the clarity, use, definition, and value of words. Of curious note is that all members are given a green costume, including a cape, feathered hat, and épée. While anyone may become a candidate for membership following the death of a member, the Académie tries to keep a balance of members from various walks of life. Writing a letter to the Head Secretary is enough to establish candidacy, but then the individual must also present themself to one or more existing members. Typically, they offer to visit each person - some accept, and some decline - and must make at least twenty. If approved, the election is not confirmed until the President, who is the Académie's protector, confirms the decision. The first woman at the Académie, Marguerite Yourcenar, was elected in 1980; in 1999 Hélène Carrère d’Encausse became the first female Head Secretary.

Source
All of the information in this writeup is from the Académie's website, http://www.academie-francaise.fr/. As one might expect, the site is entirely in French. I have tried to use my own words in most of this; some sentences are simply translated from the site.