The Citadel is the seat of the Order of Maesters, the learned men who advise the lords of the Seven Kingdoms on matters scientific, medical, and historical. It is located in the city of Oldtown in the far southwest of the continent.
The Citadel was founded many centuries ago to promote and increase learning and knowledge in Westeros. It is administered by the Conclave, the ruling council of archmaesters. The Citadel is also where young men from the entire continent are trained to become Maesters.
At Castle Black, Maester Aemon notes to Tyrion Lannister that the current summer has lasted nine years, and that reports from the Citadel say the days are growing shorter, indicating that its end is soon at hand. When Aemon later reveals his past to Jon Snow, the surprised Jon says "You're Aemon Targaryen!". However, Aemon corrects him by stating that he is "a Maester of the Citadel, bound in service to Castle Black and the Night's Watch", stressing to Jon that a sworn brother of the Night's Watch abandons all previous family ties.
A special white messenger-raven arrives at the Red Keep in King's Landing from the Citadel. Grand Maester Pycelle presents it before Queen Regent Cersei Lannister and the Small Council, explaining that it signifies that the Citadel has analyzed the climate and officially declared the ten year long summer to have finally ended, and autumn has arrived in Westeros.
When Jaime Lannister is being treated by Qyburn at Harrenhall after losing his hand, he notes that Qyburn is not a maester, as he isn't wearing a maester's chain of office. Qyburn explains that the Citadel stripped him of his chain, expelling him from the Order of Maesters, because they found some of his experiments "too bold".
In the books
In the A Song of Ice and Fire novels, the Citadel is described as a university-like complex of buildings extending along both banks of the River Honeywine in Oldtown, linked by graceful bridges. The Isle of Ravens is the oldest part of the Citadel and the site of the ravenry, from where the maesters send messenger ravens out to all parts of the Seven Kingdoms. Other locations of note in the Citadel include the Seneschal's Court, Weeping Dock, and Scribe's Hearth.