The Curator

The Curator was an elderly incarnation of the Doctor, whom had retired from his adventurous ways to become "a humble curator." He curated the Under-Gallery, where he watched over the painting Gallifrey Falls No More, after acquiring it under "remarkable circumstances."

Foreshadowing
When the Eleventh Doctor's timeline became distorted, the Curator was seen as one of his future incarnations. (COMIC: The Then and the Now)

Getting a painting
Under "remarkable circumstances," the Curator was able to acquire Gallifrey Falls No More and brought it to the Under-Gallery. (TV: The Day of the Doctor)

Interaction with his eleventh incarnation
When his war, tenth and eleventh incarnations rendezvoused at the National Gallery after the final battle of the Last Great Time War, the Curator approached Clara Oswald, telling her he was looking for the Doctor.

When the Eleventh Doctor was alone, the Curator made his presence known when the Doctor noted how he could retire and be a curator, with the Curator exclaiming that he "really [thought] [the Doctor] might." After the Doctor noted that he "never [forgot] a face", the Curator merely noted that the Doctor would revisit a few old favourite faces in years to come.

He told the Doctor that the name of the painting was neither No More nor Gallifrey Falls, but in fact one title; Gallifrey Falls No More. Despite proclaiming to be a "humble curator [who] wouldn't really know", he told the Doctor that Gallifrey was lost, and pointed him in the direction to search for it, telling him he had "a lot to do" and congratulated the Doctor.

When the Doctor tried to pry into his meaning, the Curator retaliated by pointing out it was the Doctor's choice. He was then about to tell the Doctor what he would do if he was him, but then said that "perhaps [he] was [him]," or that maybe the Doctor had been him, before deciding that it "[didn't] matter either way" and left. (TV: The Day of the Doctor)

Appearance
Having the appearance of an elderly Fourth Doctor, the Curator's face was wrinkled with baggy cheeks and a pronounced throat. He had light blue eyes and big ears with dangling ear lobes. Though he was balding, his hair was completely white, and his eyebrows were non-existent. (TV: The Day of the Doctor)

Clothing
The Curator wore a dark olive green blazer, a checked white shirt, mahogany red trousers and brown slip-on shoes with white socks. In his blazer breast pocket, he kept a red handkerchief. He also walked with a cane and wore a blue cravat under his shirt. (TV: The Day of the Doctor)

Behind the scenes
Conservateur (The Day of the Doctor) Curador (The Day of the Doctor)
 * The script for The Day of the Doctor never directly states that this character is the Doctor, but hints that he may be a future incarnation who has adopted the same form as the Fourth Doctor. The Then and the Now later depitcs him as a future incarnation of the Doctor when the Doctor's timeline becomes distorted.
 * Tom Baker is not separately credited for this performance, receiving only a credit for playing "the Doctor" (as did all the other past Doctor actors, who appeared in the episode through the use of archive footage).
 * In Doctor Who Magazine issue 487, Showrunner Steven Moffat confirms in the magazine's regular "Ask Steven" feature that the Curator is intended to be a future incarnation of the Doctor and suggests that, just as the War and Tenth Doctors retain no memory of the events of The Day of the Doctor, the Eleventh Doctor would not remember the specifics of his conversation with the Curator but will leave the Under-Gallery "with the strange, groundless conviction that Gallifrey is still out there". However as with many revelations made in this feature, and by Moffat's own admission, this is not an in-universe explanation for the character until such a time as it appears on the screen.
 * Tom Baker played a similar nameless character, with the appearance of an older Fourth Doctor, in a series of idents for the 1999 Doctor Who Night. Like the Curator, he is never stated to be the Doctor and refers to him in the third-person, but is shown in the Doctor's TARDIS and says that he's "called Paul McGann in this one" when introducing the TV movie. He also similarly appeared as the narrator for the unfinished links of Shada, speaking as the Doctor in an exhibition of the monsters he fought while also acknowledging the actors and Shada's cancellation.