The Professor was a Time Lord, and a contemporary of The Doctor. Little is known about him now, as the only record of him left is a hologram that remains as the 'autopilot' and guardian of his TARDIS, made as a complete 'mental upload copy' of him in the late 1980s. As such, the hologram's memories stop there, so everything that happened to the actual Time Lord after that is only vaguely known.
After the Five discovered his TARDIS, The Professor's hologram aided them as a tutor and guide as they grew and learned as Time Lords, slowly uncovering what had happened to his living body. Over the course of their first few months together, it was discovered that The Professor's body had been overcome with heavy doses of the Dalek Factor after having been captured by Davros and his army during the Last Great Time War. The living Professor became a far, far different person, first becoming "The Engineer", then becoming Agent Sigma, the leader and founder of the Agents of Destiny.
At the end of the Time Lord Civil War, The Professor's living body (in the form of Agent Sigma) was paralyzed by The Doctor, who gave the holographic Professor a chance to overwrite the "corrupted" Professor's bio-energy signature with his own, as Agent Sigma was considered to be "stealing" the rightful body of The Professor. The hologram agreed and forced Sigma to regenerate, using his energies to re-shape his body into what it looked like before the Last Great Time War-- copying his holographic appearance. With a new, living body at his disposal, The Professor lead the Five into the darkest reaches of the Void to continue their training.
As a boy and young man, The Professor attended the Prydonian Academy alongside The Doctor, a classmate in many of the same subjects. The two had a casual friendship though both were members of The Deca, a clique of academy students known for their rebellious behavior but outstanding abilities. Their Academy names are also quite similar, The Professor's being "Aquila Sigma" and The Doctor's being "Theta Sigma", suggesting they enrolled at approximately the same time, or were inspired by the same source. As 'young men', the two began to have divergent opinions and philosophies as to their craft, and maintained more of a professional, if a bit cold, relationship.
As a boy, he was brought to the Untempered Schism like all Time Lord candidates. He later told the Five that, "it made me feel small. It really did make me feel my place in the universe. I was scared, sad... but also awed."
Little else is known about his early life though the Doctor had this to say about him when he was mentioned: "He needs to get out of his laboratory and out of his classroom and actually live history. You can't accomplish everything inside some lab, you must feel the worlds beneath your feet!"
Life as a Time Lord
As with much of his life, little is known about the bulk of his service as a Time Lord. Apparently some time after The Doctor's exile from Gallifrey, The Professor actually did go out and start exploring the cosmos. He visited many places, so much so that he actually began to collect trophies from his adventures. Among these relics were a Glittergun (suggesting that he had fought Cybermen), a Terraforming Glove, a mobile emitter for holograms, and the Chest of Dreams, which had (supposedly) once contained a "demon".
The Professor's Companion
Sometime in his journeys, he came into the company of a young woman named Zoe. Her exact race is unknown, but she is supposed to be human, as her appearance is such. She had her own bedroom on the TARDIS, which she decorated in a soft, feminine way that was a starker contrast to the rest of the ship.
It is unclear how long they traveled together, but they did fall in love and begin a relationship together. She was quite intelligent and assisted The Professor in many of his endeavors, including the creation of a new species, which he called her the 'mother' of. This new race was often plagued by terrifying, psychic insectoids who saw his creations as an affront. The two were very close to exterminating the insects when one of their queens broke free of her bonds inside his TARDIS, where she had been kept for study. The insect showed signs of rapid learning and development, and it attempted to hijack the power of the TARDIS. Zoe, fearing for the race she had helped to create, cornered the creature and forced the console it was tapping into to overload. She locked The Professor out of the room on purpose, preventing him from stopping her. She died along with the queen, causing him to stay away from the species as for several decades.
The race that The Professor and Zoe had created were classified as 'felis sapiens', though colloquially named after their word for their homeworld, M'haura. Their creation was sanctioned by the High Council of Gallifrey, and overseen at every important step by the couple, going forward at various stages in their species evolution using the TARDIS to guide and influence it. Their appearance was humanoid, but with distinctly feline characteristics, a cosmic 'what if' that supposed if felines had been ascendant instead of apes.
They were a superstitious people at heart and had several religions, all of which were taken very seriously. While there were few actual religious wars, different interpretations of the creation of the world through "The Gates of Heaven" (an analogue of the Professor's TARDIS) led to a few factions, not the least of which was the "Sisterhood". This was considered to be the Professor's favored faction, which also made them a target in the future for the insects when they later returned to M'Haura.
After Zoe's death, The Professor ended up not visiting M'Haura for quite some time, mostly due to grief, though he did become involved in other matters. During this extended absense, the planet was visited by Doctor Ross, a Dalek/Human hybrid thought to be killed by the Five. He had stolen a passing spaceship and used it to abduct humans from Earth and other places, then used their genetic material to introduce humans to M'haura, allowing humans and felines to live together in a bizzare peace. In truth his motives were far from benign; he was using both in further genetic experiments, planning to use the cunningness of man with the strength and resilience of the M'haurans as well as the technology of the Daleks to make a perfect warrior race. The Five, along with the then-Hologram Professor, arrived on M'Haura and killed Ross (again), then carefully removed the insects from the planet, which Ross had placed there again as a means of controlling the people. The Agents of Destiny entered into a tenuous truce with the Five, helping by way of removing the humans safely back to Earth. (G5: Of Gods and Freedom, Genesis and Genocide)
The M'haurans would later go on to become major players in Galactic society.
The Last Great Time War
The Professor was present at the Cruciform, and he helped to maintenance the forward batteries which held the Dalek fleet at bay. The Master, who was acknowledged for his tactical prowess, was given command of the forward batteries and held an apt defense of the Cruciform until the Dalek fleet decided to charge the massive structure.
Since The Professor had been ordered by the Council to keep an eye on his former classmate, he went with The Master on a strategic retreat from the forward areas of the station to the engineering section, where he lead a valiant effort to keep the engines running, giving the rest of the crew time to escape while the Daleks were boarding. The Master, sensing imminent defeat, fled in an escape pod, leaving The Professor alone to try and keep the power running. When the Daleks finally boarded the Engineering section, he had planned to detonate the massive Time Rotor that powered the Cruciform, but he was captured before he had the chance to set off the explosion.
Shortly thereafter, the Dalek fleet lead by Davros, attempting a sneak attack on Gallifrey by going past the Gates of Elysium, was routed by The Doctor and his fleet, with the help of the Five and their newly-acquired Type-113 TARDIS. Rather than be captured by his greatest rival, Davros attempted suicide, flying into the Nightmare Child, a sapient black hole that sat near the Gates of Elysium. Dalek Caan, traveling through the time locks, arrived on Davros's ship in an attempt to save his creator, but his emergency temporal shift drive was damaged. They needed an engineer good with time-travel technology to repair him, and luckily they had just captured one.
The Professor was tight-beam Trans-Matted to the ship, and kept under heavy Dalek restraints, even so far as to put tubes into his body pumping constant amounts of Dalek Factor into his body. Enslaved by the Daleks, he was forced to repair Caan's timeshift device, until the Five arrived to try and stop them. A firefight ensued, and Caan opened his shell in a panic to try and escape. The Professor managed to break his hypnosis long enough to rip out one of the cables from his body and plunge it into Caan's instead, for reasons as yet unknown. Davros decided he'd had enough and ordered Caan to activate his temporal shift, carrying himself, Davros, and The Professor off into the Time Vortex.
Life as The Engineer
The trio slid out of the Time Vortex somewhere in the vicinity of 2009, where they set about building a new Dalek Empire. The Professor's body had since been corrupted by the Dalek Factor, with massive amounts of it now coursing through his body. He began life anew, his mind now being forcibly swayed to the Dalek way of thinking, and took on a new identity as "The Engineer", helping Davros to build the Crucible, and even to amass the new Dalek Army. It was also at this time that he began to develop a philosophy of unification between Time Lord and Dalek aspects, feeling the Dalek focus and dedication and perseverance in his own mind and body, yet also feeling the scope, wisdom, and tenacity of the Time Lords still lingering in his mind. The Engineer began to slowly break free of Davros's control, eventually allying himself with the new Supreme Dalek they had constructed, working with him to isolate and later imprison Davros within his own Crucible. As a reward, The Engineer was given some measure of autonomy, but not before being used as a tool to steal the plans for the Reality Engine from the TARDIS that the holographic Professor and his students were riding on.
The Creation of the Agents of Destiny
While Davros busied himself with the Reality Bomb, The Engineer focused instead on trying to create a new breed of half-Time-Lord, half-Daleks, which he reasoned would be the most unstoppable force in the Universe, able of preserving peace and justice with an iron fist. Using stolen SIDRATs and various means of espionage, he learned of the Five and began to study them at length. Even though they were impure samples of Gallifreyan DNA, it was "better than nothing" in his eyes, so he set about collecting samples of their genetic material after channeling them to his "home" deep within the Crucible during their visit. It was also at this time that he used an obscure wartime override code to take control of Harmony, the experimental Type-113 War TARDIS that the Five had claimed as their visit to the Last Great Time War. Taking her captive, he escaped the destruction of the Crucible at the hands of the Five, The Doctor, and Donna Noble.
With Harmony-- now renamed Dissonance, or "Dizzy"-- at his side, he changed his persona yet again, this time, to use a set of Gallifreyan ruins he had discovered on his fact-finding missions for the Daleks, using it as a starting point for his conquest.
Life as Agent Sigma
After creating his new persona as Agent Sigma, he set about the task of creating his half-breed experiment, which he called the Agents of Destiny, meant to preserve the timeline and preserve history at any cost. One of his greatest discoveries while exploring for the Daleks had been an old Gallifreyan Loom, which he used with his altered DNA samples to create the original Lettered Agents. Inserting himself as Agent Sigma amongst the other young Agents, he "discovered" a series of ruins meant to weave an elegant mythology for them, explaining their origins as having been descended directly from the Time Lords, and making them rightful heirs to their empire and ultimate quest and purpose.
Sigma further skewed things by placing one of the Lettered Agents, Omega, in charge of the Agency, thus putting focus off of himself, allowing him to rule from the shadows. By subtly manipulating Omega, he instructed the Agents to start collecting various Time Lord artifacts, eventually culminating in the master plan to rebuild Gallifrey altogether by finding its component parts, strewn across various parts of the Universe, and physically reuniting them into a planet again. This plan went largely unnoticed until the Agency forcibly seized Flat Earth from its residents, the floating continent actually an important missing piece of the planet.
Part of the plan to salvage Time Lord technology involved restoring several old War TARDISes that had been set adrift during the Last Great Time War. Sigma controlled this effort from the shadows as well, turning them from TARDISes into "Timeships", each of the ships now receiving traditional names and the prefix "ATS", for "Agency TimeShip". This soon became the war fleet by which the Agency enforced its will, able to go anywhere and anywhen in the Universe.
Life as a Hologram
After being reawakened by the Five, The Professor has done his best to act as a teacher to them, educating them of their Time Lord heritage. He has given some of them training on how to fly the TARDIS and has given them all some lessons on how to construct and repair a Sonic Screwdriver.
During their first mission in time The Professor's mobile emitter was destroyed by the Prometheans, and was almost stuck inside the TARDIS for good until he thought to outfit a mobile emitter onto a metalroach that the Five had domesticated, which they called BUG (Biology Underfoot & Gilded). With BUG's assistance, he can now project himself anywhere he wishes, though he must upload his program into new areas if he wants to leave the TARDIS.
While he is uploaded into BUG's matrix, the two are apparently symbiotic to a degree; at Woodstock, it was discovered that alcohol poured onto BUG's catalytic converter produced an effect in his processors that resulted in floating-point decimals in their programs-- which expressed itself a lot like the effects of alcohol.
Age and Regeneration
The Professor's age has yet to be determined. If he is a contemporary of The Doctor, that places his age at around 900 years, but the measurements get more difficult if you consider this to be an actual number of years lived as opposed to the number of tears since his birth. It is quite possible that the Doctor has nearly reached a millennium of life because he has traveled so much; it is established that The Professor did not travel much before being given his ship and started taking assignments from the High Council.
His regeneration is also difficult to place. The Doctor is already well into his 10th life because he tends to live a more dangerous existence. The Professor lived much of his time in a very 'safe' pattern until he started exploring, meaning that he could well be in very low numbers. We can safely assume it is not his first regeneration, considering how long ago his 'youth' back on Gallifrey occurred. If all of his regenerations died of old age rather than something else, he could safely be in his third or fourth regeneration.
Pessimists would argue that since he is most probably dead, any argument here would be moot.