The Conclave meets in Oldtown to discuss the changing weather patterns. They agree that summer has ended and that autumn has arrived. As is traditional, they send this news to all corners of Westeros with special white ravens. Grand Maester Pycelle relays this news to the small council, prompting them to start tallying the supplies and stores the people of the Seven Kingdoms have put aside for the coming winter.
The Conclave members read Bran's message warning of the army of the dead, but are dismissive of his account. Sam, present only to swap out some books, vouches for Bran and tells the Maesters that Bran spent several years surviving in the wilds alone. Archmaester Sandhu mockingly tells Sam to practice with inscribing instead of entertaining myths and fables. Sam counters that they should use their position as Maesters to warn the people to prepare for the coming Night. Archmaester Ebrose opines that Bran's message could be genuine, but could be disinformation spread by Daenerys. The Maesters agree to send a letter to Maester Wolkan at Winterfell to investigate Bran's claims further, but as they compare them to the works of Jenny of Oldstones and Lodos, they are skeptical. After Sam leaves, one of the other Archmaesters asks if it is true that Sam's father and brother were recently burned alive. Ebrose confirms this, and admits that he hasn't had the heart to tell Sam yet.
In the books
In the A Song of Ice and Fire novels, the Conclave is made up of the ruling archmaesters of the Citadel. In addition to discussing the change of seasons the Conclave also elects the new Grand Maester after the passing of the old, decides on policies, and attends to the administration of the Citadel as well as the training of new recruits. The Conclave also appoints the Seneschal, the maester who has to run the bureaucracy of the Citadel on a day-to-day basis, which is considered drudgery.
Much like the professors in a university faculty, each of the archmaesters is the foremost expert in a specific field, such as the archmaester of medicine or the archmaester of economics, etc. This is in contrast with the ruling council of the Faith of the Seven, the Most Devout, whose members have no official distinction between themselves. While each archmaester is thus not interchangeable, in theory each archmaester is equal, each casting one vote when electing a new Grand Maester. In practice, of course, archmaesters of certain areas of study are considered more prestigious than others. For example, the archmaesters of medicine or economics are held in high esteem, while archmaesters of obscure topics such as magic tend to be ostracized.