The policy of non-interference was fundamental to Time Lord attitudes towards and dealings with the rest of the universe.
Origin
Following the Eternal War, (TV: State of Decay) the Time Lords abandoned the use of violence, but still continued to involve themselves in the affairs of the wider universe. This had catastrophic results for the people of the planet Minyos, who regarded the Time Lords as god-like beings. However, the Time Lords gave the Minyans technology they did not have the cultural maturity to use responsibly, and nuclear war broke out, nearly putting the Minyans to extinction. (TV: Underworld) Around the same time, Time Lord meddling on Klist accidentally reversed evolution for the inhabitants there, and the residents of Plastrodus 14 were all driven mad. (PROSE: The Quantum Archangel)
Horrified by the results of their meddling, the Time Lords adopted a strict policy whereby they would only observe the events of the wider universe, and never become involved personally. (TV: Underworld) By another account, Rassilon wrote down his principles of non-intervention following a nightmarish vision of a monstrous imperialistic Gallifrey. (COMIC: The Final Chapter)
This rule was ingrained into the brains of all Time Lords. Challenging this conditioning could result in memory loss, shutting down whole aspects of personality, and madness. (PROSE: Time and Relative)
While trapped on earth in 1976, the Time Lord Marnal wrote about the policy in his novel The Hand of Time. He explained that a Time Lord's influence in reality actually shaped it, freezing potential and possibility into certainty. He asserted the universe remained in perfect fluxing balance, any and all outcomes equally possible, as long as the Time Lords merely observed and never interfered. (PROSE: The Gallifrey Chronicles)
Exceptions
Like many of their principles, the Time Lords' attitude to the non-interference policy was somewhat flexible, even to the point of hypocrisy. During the Millennium War, Rassilon broke the policy and joined the fight against the Mad Mind of Bophemeral. (PROSE: The Quantum Archangel) Before the Doctor's departure he was able to make the Time Lords implement a ban on miniscopes, an act which earned him great respect in the wider universe. (TV: Carnival of Monsters) The Time Lord Irving Braxiatel organised the Armageddon Convention, which led to the banning of many weapons of mass destruction amongst leading civilised species. (PROSE: The Empire of Glass)
The Time Lords later breached the policy on a major scale when beings from the constellation of Andromeda were able to infiltrate the Matrix and steal its secrets. The Time Lords devastated the planet and moved it light-years across space to conceal what they had done and prevent the secrets reaching Andromeda. They also manipulated events and individuals on Thoros Beta to prevent Crozier from discovering an effective means of immortality. The Doctor eventually discovered what had happened to Earth but was unaware of the reasons. The Time Lords put him on trial for his life, the initial pretext being (ironically enough) his repeated breaches of the policy. (TV: The Mysterious Planet, Mindwarp)
A major breach of the policy occurred when the Eleventh Doctor was about to die on Trenzalore. The Doctor and his companion Clara Oswald had witnessed a future where the Siege of Trenzalore had ended with the planet as a barren wasteland that was a battlefield graveyard for all who had died there, including the Doctor himself. (TV: The Name of the Doctor) Having run out of regenerations, the Doctor was on the verge of dying both from old age or Daleks extermination when Clara pleaded with the Time Lords through a crack in the universe that connected to the pocket universe in which Gallifrey was trapped in for the Time Lords to save the Doctor. In response, the Time Lords sent the Doctor a new regeneration cycle through the crack, enabling him to change the future. The Doctor proceeded to use his regeneration energy to obliterate the Daleks, saving Trenzalore from being destroyed. Furthermore, the new regeneration cycle bestowed upon him by the Time Lords allowed the Doctor to regenerate into the Twelfth Doctor, changing his personal future and relegating the Trenzalore witnessed by the Doctor and Clara to an alternate timeline. (TV: The Time of the Doctor)
Individual attitudes to the policy varied. Rodan appeared to regard it as inviolable, (TV: The Invasion of Time) whereas Romana I accepted being asked to leave Gallifrey and assist the Doctor. (TV: The Ribos Operation)
The Doctor
On occasion, the Doctor was used by the Time Lords to do their bidding for them. As the Doctor was a renegade, the Time Lords could deny they had any influence on the Doctor's actions. (TV: The Two Doctors)
It was partly the non-interference policy which led to the First Doctor leaving Gallifrey. He felt that there were evils which should be fought, and positive ways of interfering. Additionally, he claimed he was bored at home and wanted to experience the universe first hand. When the Time Lords eventually located him in his second incarnation, he was put on trial for breaching the policy.
In defending his actions, the Doctor managed to convince the Time Lords that there was some merit to his argument of justifiable interference. An ironic result of this was that the Time Lords punished the Doctor for repeatedly involving himself in the affairs of less-advanced planets by involving him in the affairs of a less-advanced planet for an extended period of time: they exiled him to Earth in the 20th century with his TARDIS rendered inoperative. (TV: The War Games)
The Time Lords repeatedly demonstrated a willingness to involve themselves in the affairs of the universe, using the Doctor as an unofficial and often unwilling agent. For a considerable period of time they used his talents to further their own agenda. For instance, they sent the Third Doctor to Uxarieus to stop the Master gaining control of the Doomsday Weapon (TV: Colony in Space) and to Solos to help the natives complete their natural metamorphosis. (TV: The Mutants) All of these occurred prior to the formal lifting of the Doctor's exile. (TV: The Three Doctors)
At some point the Time Lords installed a remote operation module in the Doctor's TARDIS, allowing them to surreptitiously control where the TARDIS went. They found the Doctor would make the right decision when he believed he was there by choice. (AUDIO: No Place Like Home)
A Time Lord appeared to the Fourth Doctor in person at the start of his mission to Skaro to avert the creation of the Daleks (a mission to ultimately have dire consequences for the Time Lords). (TV: Genesis of the Daleks)
The Fourth Doctor believed his arrival on Karn was the result of Time Lord interference. (TV: The Brain of Morbius)
Much later, the Sixth Doctor believed his involvement in the Cybermen's attempt to destroy Earth in 1985 was a result of Time Lord machinations. (TV: Attack of the Cybermen)
Last Great Time War
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The Time Lords, in order to survive the Last Great Time War, decided to make the biggest break of the policy: destroy the universe in order to survive as beings of pure consciousness, a plan devised by Rassilon himself. The Tenth Doctor stopped them with the help of his companion Wilfred Mott and the Saxon Master. In this instance, the Doctor banished the Time Lords back into the War to die and also prevented numerous other threats from coming through that would have turned the universe into hell, as the Time War was. (TV: The End of Time) However, Gallifrey was never destroyed, only frozen in a moment and placed in another dimension. (TV: The Day of the Doctor)
Other non-interference policies
Graham O'Brien asked the Thirteenth Doctor if she had a rule against meddling in other civilisations, which he thought was called the Prime Directive. Ryan Sinclair told him that he was thinking of Star Trek. (PROSE: The Secret in Vault 13)