A prophecy was a prediction of future events, usually offered by beings with precognitive abilities. Many prophecies were made by those with psychic abilities.
The Sibyl
The Sibyl foretold that the Doctor's TARDIS, the "blue box", would appear at the time of "storms and fire and betrayal". This came true in 79 Pompeii when the Tenth Doctor arrived and caused Vesuvius to erupt, saving the Earth at the cost of 20,000 lives. (TV: The Fires of Pompeii)
The Gift of Pompeii
While in Pompeii, the Tenth Doctor and Donna Noble encountered Evelina and Lucius Petrus Dextrus, both of whom possessed telepathic abilities and the ability to see into the future as a result of the Pyroviles and a time rift cracked open by Vesuvius' eruption. Because of this, they made some startling prophecies about the Doctor and Donna's futures. When asked by her parents if she could see anything, Evelina predicted that someone would have to make a terrible choice. This was about the Doctor having to choose whether to destroy Pompeii by causing Vesuvius to erupt to save the world. A prediction of Lucius' was to the Doctor: "She is returning." (TV: The Fires of Pompeii) This referred to to Rose Tyler's return. (TV: Turn Left, The Stolen Earth) He predicted that Donna had "something on [her] back". (TV: The Fires of Pompeii) This referred to a Time Beetle which attached itself to her back, changing the timeline. (TV: Turn Left)
The "false prophet" Donna Noble
The time traveller Donna Noble, who was later referred to as a "false prophet" by the Sibylline Sisterhood, prophesied to Evelina that Vesuvius was going to "explode" on 24 August 79 and Pompeii would get buried under rocks and ash. She knew this not because of any psychic abilities, but due to her historical knowledge. The Sisterhood were unable to foresee this, as the time rift that had awoken their powers of sight "echoed into the Pyrovillian alternative", but the "prophecy" nonetheless came true. (TV: The Fires of Pompeii)
Dalek Caan
When he travelled into the Time War, Dalek Caan saw all of time and space, past, present and future. The experience drove him insane, but it also caused him to work to orchestrate the Daleks' downfall. Dalek Caan predicted that the Doctor was coming and referred to him as the "three-fold man". (TV: The Stolen Earth) This referred to the three Doctors who converged to stop the Daleks, the tenth incarnation, the meta-crisis Doctor and the DoctorDonna. Dalek Caan predicted that the Doctor's Children of Time would gather and the most faithful one would die. He predicted this repeatedly. His last words were, "One will still die", as his final prophecy. In actuality, this referred to an aspect of Donna Noble, the "Doctor Donna", which the Doctor had to wipe from her mind to save her life. Dalek Caan also predicted that the Doctor's true soul would be revealed. Davros claimed that when the Doctor's companions threatened mass destruction to stop the Daleks, the truth about the Doctor was revealed; he refused to use a weapon, but moulded his companions into them. Dalek Caan predicted the end of the Dalek race and foresaw that the Doctor would be the one to cause it. Dalek Caan urged the meta-crisis Doctor to fulfil the prophecy of the Daleks' destruction and made him realise it needed to be done. At Dalek Caan's insistence, the meta-crisis Doctor destroyed the Daleks. (TV: Journey's End)
Carmen
As she said goodbye to the Tenth Doctor, Carmen made a prophecy about his future, telling him, "Your song is ending sir... It is returning. It is returning through the dark and then Doctor, oh, but then... He will knock four times." (TV: Planet of the Dead) The Tenth Doctor did everything he could to stop this prophecy from happening, even electrocuting Andy Stone to prevent him from knocking four times, (TV: The Waters of Mars) but it proved inevitable. When the Saxon Master brought back the Time Lords, the Doctor realised what "it" was. As a result of breaking the time lock on the Time War, the Master had brought back more than the Time Lords; he had brought back Gallifrey, the "it" from Carmen's prophecy that was returning. Gallifrey returned close to Earth and nearly destroyed the planet. After the Doctor and the Master defeated the Time Lords and banished Gallifrey and them back into the Time War, the Doctor believed that he was safe and had averted the rest of the prophecy, when four knocks sounded. "He" had referred to the Doctor's companion, Wilfred Mott, who knocked four times repeatedly to get the Doctor's attention to be set free from a radiation booth he was trapped in after freeing a man. To save him, the Doctor sacrificed himself and absorbed a fatal dose of radiation lest Wilfred absorb it and die. This caused him to regenerate. Thus the tenth incarnation died and the eleventh was born. (TV: The End of Time)
The Ood
On the Ood-Sphere, Ood Sigma told the Doctor that his song "must end" soon. (TV: Planet of the Ood) When the Doctor later returned to the Ood-Sphere, the Elder Ood chanted a prophecy, "It is returning. They are returning. He is returning. Both he, it and they are returning, but too late." "He" turned out to be the Master, who was brought back to life. The Master transplanted himself into every human on Earth. When all the new Masters concentrated on the drumming inside his head, he pinpointed its origin. This meant that the High Council of Time Lords, who retroactively implanted the signal in his head when he looked into the Untempered Schism, were able to send a Whitepoint Star to Earth, making contact physical. This was the "they". Along with them, Gallifrey also came through, the "it". (TV: The End of Time)
The Trickster
When the Doctor encountered the Trickster, he was told, "The Gate is waiting for you". (TV: The Wedding of Sarah Jane Smith) The "gate" was the Immortality Gate, which flooded the Doctor's body with Vinvocci radiation, forcing him to regenerate. (TV: The End of Time)
"Silence will fall"
The religious order known as the Silence believed that "Silence will fall when the Question is asked", better translated as "Silence must fall". According to Dorium Maldovar, "on the fields of Trenzalore, at the Fall of the Eleventh, when no creature can speak falsely or fail to answer, a question will be asked. A question that must never ever be answered: Doctor who?" The Silence wished to prevent the Doctor — the one man, they believed who knew the answer — from responding. They thus sought to kill him several times during his eleventh incarnation. (TV: The Wedding of River Song)
Eventually it was revealed that Trenzalore was the location that the Time Lords were attempting to return to the universe through a crack in time, broadcasting the question "Doctor who" through it. If the Doctor said his name, they would use the crack to return to the universe. The town the crack opened on, Christmas, had a Truth Field surrounding it, preventing anyone from lying. The Eleventh Doctor refused to say his name as it would cause all of the universe's most dangerous species to descend on Trenzalore and cause a new Time War, thus destroying the universe. The Church of the Silence became determined to stop this, to help the Doctor and keep silent and protect the planet from his enemies, declaring that "silence will fall." The Kovarian Chapter broke off and tried to use time travel to prevent the Doctor from ever reaching Trenzalore, causing the events that brought him there in the first place. The Doctor spent centuries defending the planet until all but the Daleks left or were destroyed. At the point of death, the Time Lords intervened by granting the Doctor a new cycle of regeneration and then closing the crack. The Doctor destroyed the Dalek ships with his regeneration energy and then regenerated into the Twelfth Doctor. (TV: The Time of the Doctor)
Apollonians
The Eye of Torment was a prophecy foretold by the Apollonians. According to Sandy Tanaka, "If the body of the Sun [were] ever violated, they believe[d] that something called "the Eye of Torment" [would] open. It [would] look down on humanity and condemn [them]." (COMIC: The Eye of Torment)
The Hybrid
According to Davros, there was an ancient Gallifreyan prophecy of a great warrior creature, the Hybrid, a fusion of Time Lords and a great warrior race - speculated to be the Daleks - more powerful than either. (TV: The Witch's Familiar)