The Master could be considered the true "opposite" of the Doctor. Where the Doctor champions the cause of good, the Master sides with evil, especially when it furthers his own ends. He is perhaps the greatest mass murderer of all time and space: his victims have included the inhabitants of Atlantis and several planets wiped from existence, including Traken, Nyssa's home planet.
The Master's ultimate goal is power. He delights in deviousness and trickery, he steals ideas (and bodies) from other people. All of his schemes will in one way or another benefit him, the major benefit being the destruction of the Doctor and/or the Universe.
The Master and the Doctor were considered contemporaries, even friends during their early days on Gallifrey. They both delighted in pulling pranks at the Academy, ruining others' experiments. After a while, they left Gallifrey, but while the Doctor righted wrongs, the Master set out to create them...
The Master led a dangerous life and by the time he was captured by the Time Lords and imprisoned, (possibly on Shada) he was apparently in his twelfth incarnation. He escaped and went to Earth, where the third Doctor was exiled. The Time Lords thought it prudent to warn the Doctor and did so, just before the Master assisted the Nestene to launch a second invasion of Earth, but the Master ended up helping the Doctor repel the invasion (Terror of the Autons).
He then masqueraded as Emil Keller and brought a Mind Parasite to Earth. He used it to extract the evil from prisoners and then use it to sabotage a World Peace Conference. (The Mind of Evil)
He then brought the Axons to Earth, but became their prisoner, he escaped the Axons, then had to help the Doctor put them in a Time Loop. (The Claws of Axos)
He then left Earth and tried to gain the Doomsday Weapon, but the Doctor, sent by the Time Lords, thwarted him (Colony in Space)
Back on Earth, the Master, posing as Mr. Magister, a vicar, summoned Azal, the Daemon to life and to ask for his power. He failed and was arrested by UNIT (The Daemons).
While in prison, he tricked his warder and allied himself with the Sea Devils. The Doctor defeated them but the Master escaped in a hovercraft (The Sea Devils)
He then posed as Professor Thascales and used TOMTIT and the Kronos Crystal to summon Kronos and cause the downfall of Atlantis. Kronos wanted to destroy the Master, but the Doctor interceded on his behalf (The Time Monster)
The Master then realised that he needed some help in conquering the Universe, he unfortunately enlisted the help of the Daleks and used the Ogrons to foment a war between Earth and Draconia in the 26th century, but was thwarted by the Doctor. (Frontier in Space)
Having apparently reached his twelfth and final regeneration, the Master turned into a decaying corpse. He was dying when Goth found him on Tersurus and brought him to Gallifrey. The Master helped Goth assassinate the President and frame the fourth Doctor for the murder. But he was really after the Eye of Harmony, which he intended to use to prolong his lifespan. He was thwarted by the Doctor, but stole enough energy to prolong his life and escape (The Deadly Assassin)
Still as a decaying corpse, the Master travelled to Traken, where with his TARDIS, disguised as Melkur, attempted to assume the Keepership of Traken and turn the Union of Traken into a powerful force. Again he was thwarted by the fourth Doctor, but managed to take over the body of Tremas, Nyssa's father. The body was not that of a Time Lord, which meant that the Master would eventually have to find another body to survive. (The Keeper of Traken)
The Master, in his new body, attempted to trap the Doctor in his TARDIS, this failed, so he followed the Doctor to Logopolis where he used a sound cancelling wave to destroy Logopolis and threaten the entire Universe with entropy. He then blackmailed the Universe with the Logopolitans calculations which were intended to save it. The Doctor again thwarted the Master, but it cost him his fourth life, falling off the radio telescope at the Pharos Project (Logopolis)
Adric was then captured, and the Master used Adric's skills in mathematics to create Castrovalva, a recursive trap for the newly regenerated fifth Doctor. The plan backfired: Adric was freed and the Master trapped in the collapsing Castrovalva (Castrovalva)
Somehow, the Master escaped, but the dynomorphic generator of his TARDIS was exhausted and he was stranded in the Jurassic period. Disguised as the Arab magician Kalid, he drew one Concorde through a time contour, hoping to attract the Doctor, he did, the fifth Doctor put his TARDIS on another Concorde which the Master also drew back through time. The Master wanted the Doctor and his TARDIS to break through the barrier of Xeraphin, which the Master wanted to use in his TARDIS. The fifth Doctor defeated him again and secretly reprogrammed the Master's TARDIS to send him to Xeriphas. (Time-Flight)
On Xeriphas, he met Kamelion, a shape shifting android. He took control of Kamelion and escaped. He used Kamelion to impersonate King John in order to prevent the signing of the Magna Carta, but was again thwarted by the fifth Doctor. (The King's Demons)
When the Death Zone was reactivated, the High Council summoned the Master and offered him a full pardon plus a new life cycle if he would enter the Zone and rescue the Doctor. The Master tried to seize Rassilon's secret of immortality, but was stopped by the Brigadier and banished by Rassilon. (The Five Doctors)
He was then accidentally shrunk to doll size while experimenting with his TCE. He took over Kamelion again and travelled to Sarn, where he hoped to use the Numismaton Flame to restore his size. The fifth Doctor caused him to be trapped inside the flame and it seemed as if the Master had been vaporised. (Planet of Fire)
As always, he managed to escape and set a trap for the sixth Doctor in 19th century England. He attempted to alter history by preventing the Industrial Revolution, he ran afoul of the Rani, who was in the period conducting experiments on the locals. Both of the Master and the Rani entered an uneasy alliance, but were still outwitted by the Doctor, who sent the Rani's TARDIS with the Master and the Rani inside spinning through the vortex. (Mark of the Rani)
Yet again, the Master regained his TARDIS, and hired Glitz to gain possession of the secrets of the Matrix, which were stolen by the Andromedan Sleepers. After Glitz had failed, the Master entered the Matrix. He was concerned about competing with the Valeyard in evil, so he sent Glitz and Mel to testify at the Doctor's trial and expose the Valeyard. The Doctor and the Valeyard battled in the Matrix, and the Master hoped that they would destroy each other, the Doctor won, but the Valeyard had substituted the secrets with a Limbo Atrophier, which trapped the Master and Glitz in the Master's TARDIS (Trial of a Time Lord: The Ultimate Foe)
The Master escaped (again), this time seemingly without his TARDIS, but was trapped on the planet of the Cheetah People. He began to turn into one of them, and used their teleport abilities to escape back to Earth, where he fought the seventh Doctor. Eventually, their conflict took the Master back to the dying Cheetah Planet, where the Master reverted to savagery. (Survival)
Apparently, the Master escaped the planet, but caught the attention of the Daleks, who captured him and sentenced him to extermination. His last request was that the Doctor take his remains back to Gallifrey. Inside the Doctor's TARDIS, the "essence" of the Master escaped the casket in which it was held, entered the TARDIS' systems and brought the TARDIS to San Francisco. The Doctor regenerated again and the Master took over another body, that of Bruce, a paramedic. He intended to use the Eye of Harmony in the Doctor's TARDIS to take over the Doctor's body and remaining regenerations. The Master was defeated (yet again) by the Doctor and he fell into the Eye of Harmony. (Doctor Who)
It is not known if the Master survived.
Thanks to Jean-Marc L'officier, author of The Universal Databank for his assistance. His work provided inspiration and background information for the above text.