Fixed point in time

Fixed points in time were moments in the space-time continuum at which events were set in stone and could never, ever be changed, no matter what. The Time Lords knew which points in time were "fixed", which the Ninth Doctor said was a maddening experience to have to go through. (TV: The Parting of the Ways) This was called "the burden of the Time Lords" by the Tenth Doctor, who was trying to explain to Donna why he couldn't prevent Mount Vesuvius from erupting or save the people of Pompeii. (TV: The Fires of Pompeii)

Definition
Fixed points were events and/or individuals who had such long-standing impacts on the timeline that no one, not even Time Lords, dared interfere with their natural progression. The Doctor, free to interfere in alien invasions and save planets in most cases, could neither interfere nor interact with these fixed points, out of fear of damaging reality. Fixed points could be flexible and did not have to happen exactly the way they had in the original timeline but meddling with one could potentially result in reality falling apart. Were a fixed point to be interfered with, time would often find a way to make the timeline continue with minimal changes.

For example when the Doctor saved Adelaide Brooke and two of her crew, Brooke (to whom the Doctor had confided the nature of fixed points, and more specifically, why her death is one) committed suicide to preserve the timeline with minimal changes. Because of the survival of the other two crew members, the events that occurred on Bowie Base One were revealed to the public. Adelaide was hailed as a hero for stopping the viral menace, which did not happen in the original timeline, but this and her suicide ensured that she would be an inspiration to her descendants. On this occasion the Doctor had been able to bend a fixed point. However, he risked the safety of the whole of reality in the process as he had not truly broken it because of Adelaide's sacrifice. (TV: The Waters of Mars)

Another example is that the Doctor was fated to die at Lake Silencio, meaning he had to die there, or more precisely, that the universe needed to believe that he had died there. That didn't stop others from attempting to kill him, however, as when Gantok prepared to kill him in revenge for being beaten at live chess, the Doctor was saved from being shot by his non-destined killer by a trap hole that fed Gantok to the ravenous skulls of the Headless Monks. (TV:The Wedding of River Song)

However if one actually broke a fixed point in time, as when River Song refused to kill the Doctor, time would freeze and collapse; reality would "die". If this happened whomever had broken the fixed point had to make physical contact with the person who was also a main part of it. In other words, when River shot the Doctor at Lake Silencio, the fixed point in time was focused entirely on just the two of them and time was still in flux all around them, explaining why the Doctor couldn't get time started again by touching either Amy Pond or Rory Williams, who had also been at Lake Silencio. If the Doctor and River touched each other - or kissed as they did at their wedding - time would start moving again. It should be noted that River is the only known person to change a fixed point to such a degree that the whole of reality was put in danger.(TV:The Wedding of River Song)

Those who created a fixed point in time also had to be very specific about what they wanted to happen in it. The Silence simply assumed that the fixed point at Lake Silencio meant that the Doctor would die, never realising that he could actually fake his death instead. The Doctor did turn up for his "death", just as he was destined to do, but he had "dressed for the occasion" and was safely inside the Tesselecta at the time, fooling almost the whole Universe into thinking he was dead.(TV:The Wedding of River Song)

With a "still point in time", such as Lake Silencio, it was easier to create a fixed point in time. The only known way to actually create a fixed point in time was by writing events down. If someone read about events that were going to happen to them in the future, then the events had to happen the way they had been read because a fixed point would be created. Time was still in flux as long as the reader had not read about his or her own future. The Doctor had seen records of his death at Lake Silencio, but the Silence were never specific about whether he was going to actually die or just fake his death, which allowed the Doctor to outwit them. (TV: The Wedding of River Song, The Angels Take Manhattan)

Fixed points can even apply to living beings although there is only one known example of a living fixed point. Jack Harkness was a unique fixed point in time and space who was made immortal by the Bad Wolf. The Doctor sensed that Jack was a fixed point immediately after his first resurrection. (TV: The Parting of the Ways, Utopia)

It should be noted that the Doctor, during his first incarnation, held to a much stricter definition of fixed points, once telling his companion Barbara Wright that "not one line" of history could ever be tampered with (TV: The Aztecs), although he was being hypocritical as he had recently had extensive interaction with historical figures such as Marco Polo. (TV: Marco Polo)

Notable fixed points
Far from defining a fixed point, the First Doctor implied that the outcome of the battle of Hastings could have been changed by the Monk, but that would have put at risk the "pattern of world history". (TV: The Time Meddler)

Implied to be a fixed point was the death of Pete Tyler on 7 November 1987. Rose Tyler saved his life and the paradox allowed the invasion of Reapers. The timeline returned to normal with the Reapers gone, when Pete realised what had happened and ran out in front of the car that should have killed him. The only thing changed in the new timeline was where he died, that the driver of the car which struck him stopped and someone (Rose) was with him when he died. (TV: Father's Day)

The destruction of Ockora in 2204 was a fixed point. When the Second Doctor warned the Selachian Supreme Leader, he realised they could have done something about it. The Supreme Leader and forty Selachian soldiers escaped the planet's destruction and boarded the Triumph, the ship which dropped the G-bomb which destroyed Ockora. They killed everyone on the ship, but the other G-bomb was accidentally set off, leading to the formation of another black hole. The Doctor considered the formation of another black hole an acceptably small change. (PROSE: The Final Sanction)

The Fifth Doctor explained to Nyssa that the Great Fire of London would have happened whether they were there or not (TV: The Visitation, Black Orchid) implying it to be fixed in time. The Sixth Doctor later claimed the Mary Celeste was a fixed point. (AUDIO: The First Sontarans)

Jack Harkness became a fixed point in time after his resurrection by Rose Tyler, which caused him to become immortal. (TV: The Parting of the Ways, Utopia) Jack temporarily lost his immortality due to Miracle Day, but regained it when mortality was restored to the rest of the world. As an apparent side effect of what they'd needed to do to restore mortality, Rex Matheson was also able to return from death at least twice afterwards, however, it is not known if they both became fixed points. (TV: The New World, The Blood Line)

The destruction of Pompeii by the Vesuvius volcano was a fixed point in time caused by the Doctor. As well as its eruption, the deaths of most of the citizens of Pompeii were fixed. The Doctor did save one family, despite his unwillingness to interfere further. (TV: The Fires of Pompeii)

The death of explorer Adelaide Brooke was one of the few times the Doctor intentionally interfered with a fixed point. His rationale was that, as the last surviving Time Lord, the Laws of Time were his to command. In the end, Brooke committed suicide, allowing the timeline to unfold with only minor changes. (TV: The Waters of Mars)

The Doctor also interfered with a fixed point in time shortly before or after the previous event, by saving the life of Emily Winter, a film actress in 1920s-era Hollywood. He was put on trial by the Shadow Proclamation for this. (COMIC: Fugitive)

The Doctor's apparent death in his eleventh incarnation, was a fixed point in time arranged by the Silence. The Doctor was seemingly killed in Utah, at Lake Silencio, on 22 April 2011 at 5:02 pm. When River Song tried to prevent this, an alternate timeline was created where all of time occurred at the same time and it was always 22 April 2011, at 5:02 pm. The Doctor set things right by kissing River, shorting out the time differential between them and making events revert to the moment when she was supposed to kill him. It was later shown that the fixed point was actually not his death, and the Doctor who had "died" was actually the Teselecta - with the Doctor himself safely inside. (TV: The Impossible Astronaut, The Wedding of River Song)

While the Doctor was not against assisting friendly forces in a limited capacity, or dealing with isolated incidents (TV: The Empty Child/The Doctor Dances, et al), the overall events of the Second World War were considered fixed points to the extent that the Doctor could not support Winston Churchill's use of Daleks to accelerate Allied victory (TV: Victory of the Daleks) or Mels' attempt to back to a point prior to the war and kill Adolf Hitler (although a brief encounter with the dictator was unavoidable) (TV: Let's Kill Hitler).

Time in flux
The opposite of fixed points was time in flux. At these points time could change completely.

Flux points were relatively insignificant (on a universal scale) events that could be altered with relatively little to no consequence. The Doctor often meddled at these moments. (TV: The Christmas Invasion, PROSE: I am a Dalek) When the Tenth Doctor first met Martha Jones, he told her that "Crossing into established events is strictly forbidden ... except for cheap tricks." (TV: Smith and Jones)

Fluxing points

 * 1969 was in flux, as discovered by the First Doctor when it was revealed that WOTAN was supposed to conquer Earth (PROSE: The Time Travellers) instead of being defeated by the Doctor. (TV: The War Machines)
 * Sarah Jane Smith was shown an alternative timeline where the Earth of 1980 was a barren wasteland by the Fourth Doctor, who explained 1911 was in flux because of the threat of Sutekh, noting that history could be altered on that kind of scale when dealing with a being of Sutekh's immense power. (TV: Pyramids of Mars)
 * The Doctor explained to Rose Tyler when she said that he couldn't give dead human corpses to the Gelth, as she knew for a fact that dead bodies weren't walking around in 1869: time was in flux, changing every second and her "cosy little world can be rewritten like that". (TV: The Unquiet Dead)
 * Although the Doctor told Rose that Harriet Jones would serve three successive terms as British Prime Minister, he himself would, in his next incarnation, cause her political downfall, leading to her losing the title of Prime Minister to Harold Saxon. (TV: The Sound of Drums) She would die the following year at the hands of the Daleks during their invasion of Earth. (TV: The Stolen Earth)
 * When Edward VII, along with Balmoral Castle, vanished into thin air, the Doctor explained that with him gone, the whole future of the royal family was threatened and there would be no George V, George VI, Elizabeth II, Charles III and Camilla, William V and so on. (PROSE: Revenge of the Judoon)
 * When the Doctor and Donna were chasing Reverend Golightly to save Agatha Christie, the Doctor explained that Agatha could die and most of the books she wrote would disappear. The Doctor specificly told Donna that "time is in flux." (TV: The Unicorn and the Wasp)
 * After they saw Frank Openshaw exterminated by a Dalek, Rose Tyler and the Doctor meddled in history for him so that he met his wife Sandra years earlier than in the original timeline. (PROSE: I am a Dalek)
 * In 2020, when a drilling operation in Cwmtaff disturbed a Silurian civilisation, the Eleventh Doctor told Amy Pond, Nasreen Chaudhry and Eldane that this encounter could lead to either a peaceful relationship or a devastating war. The Doctor called the event an opportunity. (TV: Cold Blood)
 * When the Dalek Emperor and his flagship survived the end of the Time war in 199,909, they secretly altered the history of the Fourth Great and Bountiful Human Empire, controlling it via the Jagrafess. While the Ninth Doctor remembered it as the human race at its height, humanity's development was instead stunted. (TV: The Long Game, Bad Wolf)
 * At one point, the Korven altered the history of Earth (which they had previously invaded in 2480), causing part of it to be controlled by the corrupt, totalitarian Department. (TV: The Eclipse of the Korven)
 * On at least two (known) occasions, The Trickster, or someone associated with him, created alternate timelines by changing two events. On one occasion, Sarah Jane Smith died as a child (TV: Whatever Happened to Sarah Jane?), and later a member of the Trickster's Brigade facilitated an alternate timeline in which Donna Noble never met the Doctor, which resulted in his premature death, which had a domino effect leading to much of civilisation collapsing on earth and also the deaths of Sarah Jane and other associates of the Doctor (TV: Turn Left).