Rift

Time rifts, also known as temporal rifts, were weaknesses in space and time through which people or objects could pass from one location and time to another. The Seventh Doctor once explained that "rift" wasn't quite the same word, and that a rift per se was a fracture running through a four-dimensional area of space-time. (PROSE: Set Piece)

Effects
Certain rifts could transport living or non-living matter through space and time from one end to another. (TV: Adrift) The exit points of a rift could change their temporal coordinates, or move through space but not time, while other exit points could move through both space and time. An exit point could be anchored and could also emit time storms. (PROSE: Set Piece)

Individuals (TV: Image of the Fendahl, The Unquiet Dead) or even whole species, such as the Owse developed a degree of time sensitivity or other psychic powers from developing or growing up near a time rift. (PROSE: The Quantum Archangel) On some occasions, the leakage from a rift would cause things to either age or de-age, sometimes both at the same time. On other occasions, it would simply freeze time. (PROSE: ...And Eternity in an Hour)

Interactions with technology
Technology such as the Torchwood Institute's Rift Manipulator could, to a limited degree, control time rifts, but this was inexact. (TV: Captain Jack Harkness, End of Days, COMIC: Rift War!) After the Last Great Time War, the Doctor's TARDIS could use time rifts to power up. (TV: Boom Town)

On Earth
Cardiff possessed one major time rift. (TV: The Unquiet Dead, Boom Town, Everything Changes et al.) A time rift called the Dreadful Flap opened above Darlington in 1965. (PROSE: The Dreadful Flap) A small rift was located in Mexico, where John Hart collected alien artefacts. (COMIC: Shrouded) Washington DC was once about to disappear up a time rift, until this was prevented by the Seventh Doctor. (PROSE: Vampire Science)

A rift in Coal Hill Academy (TV: For Tonight We Might Die, The Coach with the Dragon Tattoo, Nightvisiting) was opened by the Shadow Kin (TV: Co-Owner of a Lonely Heart) and was mostly closed by the Twelfth Doctor. He explained that he couldn't close it entirely because the excess artron energy around the school made time itself weak there. (TV: For Tonight We Might Die) April MacLean also made a temporary rift to the the Underneath in her own backyard with a Shadow Kin scimitar, through which she and Ram Singh entered. (TV: Co-Owner of a Lonely Heart)

The eruption of Vesuvius in 79 AD briefly created a rift, echoing back in time and giving some inhabitants of the city of Pompeii precognitive powers. This rift was created because of the immense power of Vesuvius' eruption, but it opened for only a second. (TV: The Fires of Pompeii)

Two time rifts existed on Earth in ancient Egypt and 1871 Paris, being two of several fractures created by Kadiatu Lethbridge-Stewart's time travelling. The Seventh Doctor explained that these fractures could not be mended, and that they damaged the space-time they passed through. (PROSE: Set Piece)

The Eleventh Doctor described the crack in time in Amelia Pond's wall as a time rift. (TV: The Wedding of River Song)

The Picts knew of an inter-dimensional temporal rift in Aberdeen that could allow the Eaters of Light to enter Earth, albeit only one at a time. They built a cairn around it and had a "Keeper of the Gate" periodically open the portal to prevent it from widening. Whilst the portal was open, a single warrior would enter and fight the creatures back. As time passed much more slowly in the creatures' dimension, a few minutes of battle there equated to six or seven decades of peace outside. (TV: The Eaters of Light)

Little Caldwell had a time-rift in which Isaac Summerfield II and his fellow Spacefleet operatives fell through. They were transported from the 26th century to the town on November, 1963 (PROSE: Return of the Living Dad).

Elsewhere
A mobile time rift travelled through Mutter's Spiral, likely from an accident on Heracletus. (PROSE: Zeitgeist) The Third Doctor was eventually able to stop it on Alrakis. (PROSE: ...And Eternity in an Hour) Another time rift was located in a region of space near the planet Kaesov. (PROSE: The Quantum Archangel)

The Medusa Cascade once had a time rift, which was healed at some point by the Doctor, (TV: Last of the Time Lords, The Stolen Earth) after the Eighth Doctor had previously intended to use the Great Key to seal this rift in the Last Great Time War. (COMIC: The Forgotten)

The Arkheon Threshold was a time rift in the core of the planet Arkheon. (PROSE: Prisoner of the Daleks)

A time rift opened in the TARDIS itself when it was caught in a salvage ship's magnetic hobble-field. This rift bore a physical resemblance to a crack in time. This caused the near past and the near future to start leaking inside the TARDIS. The Eleventh Doctor was able to use the rift to change the timeline by travelling back in time to the moment the TARDIS was caught by the salvage ship and instruct his past counterpart on how to break free. (TV: Journey to the Centre of the TARDIS)

In E-Space, near the Gateway, there was a time rift connecting a past where the Tharils had enslaved a humanoid species and a future when their roles were the opposite. The Fourth Doctor crossed it back and forth. (TV: Warriors' Gate)