Borusa was a notable and ultimately tragic figure in Time Lord history, and a very important figure in The Doctor's life. His later life gives us some fascinating insights into Gallifrey's political and governmental processes, as well as exposing the Doctor's naivety and frequent irresponsibility in such matters.
Biography
A tutor at the the Academy during the Doctor's time there, Borusa had risen to the rank of Cardinal in the Time Lord hierarchy by the time of the Master's near-catastophic attempt to exploit the Eye of Harmony (The Deadly Assassin). The assasination of the incumbent president just prior to his resignation, followed by the exposure as a criminal and subsequent death of the leading presidential candidate (Chancellor Goth), left Borusa as the de facto head of Time Lord society pending the inauguration of a new president. However, the only remaining candidate, and thus President Elect (The Doctor), was banished (quite willingly) from Gallifrey - by Borusa - in order to help cover up the embarassing truth about Goth.
In the Doctor's absence Borusa officially became Chancellor (although the Doctor later claims this was done "illegally"), but he also remained acting president rather than staging new elections, apparently on the pretext that there was already a perfectly legitimate President Elect (The Doctor), who just unfortunately happened to always be elsewhere. In this way the Doctor inadvertently gave Borusa the keys to almost unprecedented power over the Time Lords and their technological legacy.
Although the Doctor returned to Gallifrey briefly to officially take up his presidency, this was part of a strategy to defeat the Vardans (The Invasion of Time). Once they and the Sontarans had been seen off, the Doctor - now having been officially inaugurated as President of the High Council of Time Lords - departed, again leaving Gallifrey in Borusa's hands.
At some point, Borusa officially became instated as President. He then used his presidential powers to investigate the legacy of Rassilon, including Gallifrey's "Dark Time". He hatched a plan to gain access to the Tomb of Rassilon, where he believed he would find the secret of immortality, and thus be able to rule Gallifrey forever. He reactivated the ancient Time Scoop, and revived the long forbidden war games between captured representatives of various species, in the "Death Zone" surrounding the Tomb. He summoned several of the Doctor's incarnations to the Zone, knowing that at least one of them would gain access to the tomb, and tricked them into activating the transmat that allowed him to join them there. However, the promised immortality proved to be a trap set by Rassilon to capture power-mongers such as he, and Borusa was condemned - like several others before him - to live his "immortal" life trapped within a sarcophagus. (The Five Doctors)
In further twists on the matter of the presidency, the Doctor was immediately re-elected as president by a grateful council - and fled, leaving Chancellor Flavia with full powers "until I return". He was eventually deposed, but after exposing more corruption in the High Council, resulting in an insurrection, he was offered the presidency again. He declined, but nominated The Inquisitor for the post (The Trial of a Time Lord). However, he apparently felt entitled to still refer to himself as "President of the High Council of the Time Lords" and "Keeper of the Legacy of Rassilon" after this time (Remembrance of the Daleks), suggesting that under the Gallifreyan system, while the post of president may pass to another, the title of "President" (with a capital P) is retained.