Fixed point in time

Fixed points in time were moments at which events could not change. Time Lords saw this in their minds at all times, which is maddening according to the Ninth Doctor. (DW: The Parting of the Ways) This was called 'the curse of the Time Lord' by the Tenth Doctor. (DW: 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 to interfere with their natural progression. The Doctor, while free to interfere in alien invasions and save planets in most cases, could neither interfere nor interact with these fixed points. If a fixed point were interfered with, all of time would become stuck and the universe would die. With a "still point in time", such as Lake Silencio, it was easier to create a fixed point in time. (DW: The Wedding of River Song) However, small details from the fixed points can be changed. Evidenced by Vincent still commiting suicede, but the Krafayis in his painting no longer existing and his sunflower picture being dedicated to Amy. (DW: Vincent and the Doctor)

Notable fixed points
Jack Harkness became a fixed point in time after his resurrection by Rose Tyler, which caused him to become immortal. (DW: The Parting of the Ways, Utopia) Jack later lost his immortality due to the Miracle Day phenomenon but it is unknown if this meant he was no longer a fixed point. (TW: The New World) After the Miracle ended, Jack regained his immortality, and Rex Matheson, having received a transfusion of Jack Harkness's mortal blood while rendered undying by the Miracle, and in combination with the mysterious properties of the Blessing, became immortal in the same manner as Jack, possibly indicating he is a similar fixed point in time. (TW: 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, however, did save one family despite his unwillingness to interfere further. (DW: 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 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. (DW: The Waters of Mars)

The Doctor also interfered with a fixed point in time a short while 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. (IDW: Fugitive)

The death of the Doctor himself, in his eleventh incarnation, was a fixed point in time; the Doctor was killed in Utah, at Lake Silencio, on 22nd April, 2011 at 5:02pm. When River Song tried to prevent this, an alternate timeline was created where every part of time was occuring at the same time and it was always 22nd April, 2011, at 5:02pm. The Doctor set things right by kissing River, shorting out the time differential between them and causing everything to revert to the moment where she was supposed to kill him, which she did. It was later shown that the Doctor cirumvented this by having the Teselecta "die" in his place. (DW: Let's Kill Hitler, The Wedding of River Song)

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

Flux points were relatively insignificant (on a universal scale) events that could be altered with no consequence. The Doctor often meddled at these moments. (DW: The Christmas Invasion, NSA: 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.'. (DW: Smith and Jones)

Fluxing points

 * 1969 was in flux, as discovered by the Doctor when it was revealed that WOTAN was supposed to conquer Earth (PDA: The Time Travellers) instead of being defeated by the Doctor. (DW: The War Machines)
 * Sarah Jane Smith was shown an alternative timeline by the Doctor, who explained 1911 was in flux because of the threat of Sutekh. (DW: 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". (DW: The Unquiet Dead)
 * Although the Doctor told Rose that Harriet Jones would serve three successive terms as British Prime Minister, he himself would later, in his next incarnation, cause her political downfall, leading to her losing the title of Prime Minister to Harold Saxon. (DW: The Sound of Drums) She would die the following year at the hands of the Daleks during their invasion of Earth. (DW: The Stolen Earth)
 * When the Doctor and Donna were chasing after Reverend Golightly to save Agatha Christie, the Doctor explained that Agatha could die and most of the books she wrote would disappear. (DW: 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. (NSA: I am a Dalek)
 * In 2020, when a drilling operation in Cwmtaff disturbed a Silurian civilisation, the 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. (DW: 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 to be the Human race at its height, humanity's development was instead stunted. (DW: 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. (K9TV: The Eclipse of the Korven)