The two Sashes of Rassilon were among the Artefacts of Rassilon, one of the Founders of Gallifrey. One of the two was lost early on in Gallifreyan history while the other became one of the items of office of the President of the High Council.
By the Doctor's time, the Sash of Rassilon was seen as simply a sacred item, to which the Time Lords bowed down. (TV: The Invasion of Time) However, it was in actuality a cellular manipulator which altered its wearer's biodata. (AUDIO: The Eleven, PROSE: The Legacy of Gallifrey) The wearer of the Sash was nearly indestructible, able to withstand the full power of the Eye of Harmony and control it, for example to gain a new regeneration cycle. Even in a damaged state, the Sash could be used to absorb energy from the partly-opened Eye and extend one's life, though not complete a full regeneration. (TV: The Deadly Assassin) The Sash could also be used offensively: linked to the correct biodata, it could rip Time Lords apart. (AUDIO: The Eleven)
Rassilon created two protective Sashes. One of the two Sashes was given to Omega, and lost with its owner in murky circumstances. The other, worn by Rassilon himself, (PROSE: The Scrolls of Rassilon) was passed down the generations of Presidents of Gallifrey; it was among those Artefacts which were given to a newly elected Head of the Presidency as symbols of office. (TV: The Invasion of Time)
During his term as Lord President in his first incarnation, Morbius wore the Sash of Rassilon. When he was formally deposed by his own High Council, Junior Cardinal Borusa stripped him of the Sash. (PROSE: Warmonger)
Over its aeons of disuse, the Sash became "damaged". It was unable to protect one Time Lord President from death when he was shot with a staser by Chancellor Goth on the order of the Decayed Master, who wished to use the Sash and the Eye of Harmony to give himself new regenerations as well as truly become the "Master of All Matter". Not realising that the President's death confirmed the Sash was damaged, the Master stole it from the Presidential cadaver, and began unfastening the restraints to the Eye's power.
The Sash helped him to convert some of the "tremendous energy" emanating from the obelisk, which was tearing the Capitol apart around him, but the Doctor managed to stop him from completely unchaining the power of the Eye. The Master was still wearing the damaged Sash when he fell into a chasm, a fall which he survived, soon making off in a stolen TARDIS. (TV: The Deadly Assassin)
However, a Sash of Rassilon was still among the symbols of Presidential office by the time of the Fourth Doctor's next visit to Gallifrey. (TV: The Invasion of Time) Attending the opening of a new Capitol building as Lord President, the Fifth Doctor donned the Sash of Rassilon as part of a Presidential dress he was forced to wear. (AUDIO: Time in Office)
When TARDIS repair technician Plutar was transported to Gallifrey's distant past by the reptilian humanoid Sillarc in a stolen TARDIS, Plutar found himself confronted by two Gargantosaurs and used the oath "By the Sash of Rassilon!" (COMIC: The Stolen TARDIS)
In one of the infinite parallel universes of "possible space", (COMIC: Fire and Brimstone) a singular Sash of Rassilon was the only marker of the holder of the Presidential office. (PROSE: The Chronicles of Doctor Who?)