There were countless billions of dimensions, some of them incomprehensibly different to N-Space. Beings from these distant dimensions obeyed very different laws of physics and were (presumably) forbidden to cross into other dimensions. Nobody No-One came from a reality 45 billion dimensions away from N-Space and was composed of language and communication instead of matter and energy. (AUDIO: The Word Lord)
The Sixth Doctor described there being trillions and trillions of alternate dimensions, some of which could be infinitesimally small micro-universes. (AUDIO: The Macros) However, the Seventh Doctor later commented that one billion cubic centimetres was small in terms of a dimension's size. (AUDIO: The Light at the End)
Time was often considered by humans to be the "fourth dimension" (TV: "An Unearthly Child") and Vicki Pallister later elaborated that time, even though a dimension in itself, had dimensions of its own. The First Doctor even stated, in their adventure at the time, that he did not understand "this" fourth dimension. (TV: The Space Museum) Different universes possessed different time dimensions. Time moved faster in Capron's universe, requiring metabolisms to be altered so as to prevent rapid ageing when travelling to it from N-space. (AUDIO: The Macros)
According to the Twelfth Doctor, the Boneless were "from a universe with only two dimensions", although the Doctor also described them as "creatures from another dimension" and mentioned a way of sending them "back to their own dimension". People in N-Space, where three dimensions were required for life to survive there, had theorised such a universe's existence, but according to the Doctor, "no-one could go there and prove its existence without a heck of a diet". (TV: Flatline)
"Dimension" was sometimes used interchangeably with "universe" or "plane".
Dimensions within the universe
Depending on perspective, the "normal" universe was variously described as being three dimensions, (AUDIO: Paradise 5) four dimensions, (AUDIO: Storm Warning, The Hollows of Time) and five dimensions, with the other six "higher dimensions" forming the Six-Fold-Realm.
When the universe was created, dimensions started to solidify out of the primal chaos [...] And in the aftermath of Event Zero, eleven dimensions did fight for existence. Five were triumphant - together they did become the three dimensions of space, and the two dimensions of time through which we travel. But the remaining six dimensions did still exist: although beaten, although denied their dominance, they curled and curdled amongst themselves to become a six-fold universe, separate yet conjoined.
The first three, the three "primary dimensions," (AUDIO: The Invasion of E-Space) were required to sustain native life forms such as humans and Time Lords. As the Twelfth Doctor theorised, "Perhaps [the Boneless didn't] even understand that we need[ed] three dimensions to live in." (TV: Flatline) While these made up space in the physical sense, time was considered to be the fourth dimension and space the fifth, (TV: An Unearthly Child) altogether forming the space-time continuum where space and time were connected, since they both came into being at the same moment. (AUDIO: The Forbidden Time) In relation to four-dimensional space-time, the universe had seven "concealed" dimensions, "the intermediate dimensions," which held space together. (AUDIO: The Hollows of Time)
Alternatively, access to the "fifth dimension" was equated as being access to time travel. (AUDIO: Mission to Magnus) After the bite of a Vortisaur aged the skin of Peter Rathbone's forearm to that of a sixty-year old, the Eighth Doctor described the Vortisaurs, who were native to the Time Vortex, as "five dimensional predators [who] make five dimensional bite marks". (AUDIO: Storm Warning)
According to one account, a TARDIS' interior existed as a separate dimension from the rest of the universe. This meant the interior didn't exist, creating temporal grace. (TV: The Hand of Fear) Another account described TARDISes as being a part of the universe; the Tenth Doctor said that his TARDIS was "the last TARDIS in the universe". (TV: Rise of the Cybermen) According to another account, the Eleventh Doctor said that the Doctor telling Mels that the TARDIS existed in a state of temporal grace was "a clever lie" to ensure that Mels didn't fire her gun. (TV: Let's Kill Hitler) Nonetheless, the Eleventh Doctor also described the huge room inside the TARDIS which was surrounded by a tiny exterior as "basically another dimension". (TV: The Vampires of Venice) The external dimensions of the Doctor's TARDIS were once leeched by the Boneless, making its exterior shell smaller but keeping the interior the same size. (TV: Flatline)
The 2Dis was able to restore three dimensions to small objects the Boneless had flattened into two dimensions and reverse the process. After the Boneless took on a 3D form in N-Space, they had the ability to "restore" dimensions as well. (TV: Flatline)
E-Space was a small dimension connected to N-Space by the Logopolitans in an effort to delay the death of the universe from entropy. Access was through a Charged Vacuum Emboitment. The Fourth Doctor and Romana II stumbled through one such CVE. (TV: Full Circle) They found the Great Vampire hiding there. (TV: State of Decay)
The Antiverse was a universe comprised of anti-time. Use of the Oubliette of Eternity would send people to this universe, turning them into Neverpeople. The Eighth Doctor and Romana II went there looking for the remains of Rassilon. (AUDIO: Neverland)
The Divergent Universe was a universe created by Rassilon to contain the Divergence, a race that would theoretically replace the Time Lords. The Eighth Doctor went there on a self-imposed exile. (AUDIO: Zagreus)
An empty dimension was a "spare dimension" "every TARDIS" carried as standard equipment. Someone imprisoned in there could not die, but they could be caged until the end of eternity. (COMIC: Doctor Who and the Iron Legion)
At the fall of Gallifrey, the first thirteen incarnations of the Doctor relocated Gallifrey to a "parallel pocket universe", (TV: The Day of the Doctor) or bubble universe. (TV: Spyfall) Both the Twelfth Doctor and Missy acknowledged this as "another dimension". (TV: Death in Heaven)
Species from different dimensions
Many species were either native to different dimensions or were extra-dimesnional in nature. The transcendental beings, such as the Eternals, existed in the higher dimensions of Calabi-Yau Space. The Guardians of Time (AUDIO: The Destroyer of Delights) and the Chronovores (PROSE: Blood Heat) found it "cramped" existing in the normal dimensions.
Other extra-dimensional species included the two-dimensional Boneless, (TV: Flatline) five-dimensional Tef'Aree, (COMIC: Tesseract) seven-dimensional Legions, (PROSE: Lucifer Rising) and eight-dimensional Scourge. (AUDIO: The Shadow of the Scourge) Ace claimed that the Elysium parasites, such as the Artifact, extended into more than eleven dimensions (PROSE: Parasite)
Species that were native to other dimensions included the Spillagers, (AUDIO: Winter for the Adept) Good Daleks, (AUDIO: Project Infinity) Neutron eaters, (AUDIO: Lost Souls) Edgers, (COMIC: On the Cards) and light-eating locusts. (TV: The Eaters of Light)