The Doctor's species

The Eighth Doctor claimed to Professor Wagg that he was half-human on his mother's side. (TV: Doctor Who)

Much like their age and their early life, the Doctor's species was a matter of much contention due in part to shifting timelines. (PROSE: Celestial Intervention - A Gallifreyan Noir)

The vast majority of sources agreed that the Doctor was a Gallifreyan and a Time Lord, (TV: The War Games, et al.) but a few suggested that they had different origins. Various accounts identified the Doctor as being fully or partially human, (PROSE: Doctor Who and the Daleks, TV: Doctor Who, et al.) having once been the Timeless Child from an unknown species, (TV: The Timeless Children) or as the product of still stranger origins. (PROSE: The Death of Art, Sometime Never..., et al.)

Time Lord

A scan taken by Kate Stewart reveals the Eleventh Doctor's binary vascular system. (TV: The Power of Three)

Although human-like in appearance and broad mannerisms, the Doctor was by most accounts not human; (PROSE: Who is Dr Who?) the Doctor usually identified themselves as a Time Lord (TV: The War Games, Pyramids of Mars, The Night of the Doctor, et al.) from the planet Gallifrey in the constellation of Kasterborous. (TV: The Day of the Doctor) The Fourth Doctor once explicitly referred to himself as "a native Gallifreyan Time Lord". (PROSE: A Letter from the Doctor)

A previous life, a previous race

According to one account, the First Doctor was indeed the first incarnation of the Doctor as such, but was the reincarnation of the Other, one of the Founders of Gallifrey, whose mind and genetic structure had been redistributed into the Loom which created the Doctor many millennia later. (PROSE: Lungbarrow) According to The Thousand and Second Night, a highly figurative account of early Gallifreyan history, an individual who was "neither an angel nor a djinn" was involved in the Eternal War. The Shift claimed to be them. (PROSE: Head of State)

In a similar but distinct account, however, the Doctor's childhood on Gallifrey with the Master was a result of their having been regressed into a child, after having had their memories redacted, following a large number of regenerations; they were originally the Timeless Child, a being of an unknown species whom the Shobogan explorer Tecteun adopted and brought back to Gallifrey, harnessing the Child's regenerative abilities to turn her people into the Time Lords. Though she sequenced her adopted child's entire genome, Tecteun was still no closer to discovering the true origins of the Doctor. (TV: The Timeless Children)

Half-human?

The Eighth Doctor had vivid memories of his childhood on Gallifrey, but he also claimed that his mother was human; the Bruce Master's analysis of the Doctor's retinal structure seemingly confirmed that he was half-human, (TV: Doctor Who) although the Eighth Doctor would later claim that he had tricked "his greatest enemy" into thinking him to be half-human using a "half-broken Chameleon Arch". (COMIC: The Forgotten) While the Eighth Doctor was on Dreamstone Moon, Isabella Cleomides reported that his retinal structure instead denoted Partriscisnad origins. (PROSE: Dreamstone Moon)

The Eighth Doctor, when being interrogated about his origins, also once claimed that he was from Andromeda, his mother having been "abducted by little green men". (PROSE: Seeing I)

Me briefly speculated that the prophecy of the Hybrid referred to the Doctor, believing him to possibly be half-human based on his tendency to spend time on Earth. However, the Twelfth Doctor laughed at this theory. (TV: Hell Bent)

While the Relic, the corpse of the Doctor's final incarnation, was in storage awaiting auction in the Unthinkable City, Qixotl saw Sam Jones and Kathleen Bregman on the security feed attempting to reach it. Qixotl speculated that the Relic's lingering consciousness had telepathically summoned them, as "the Doctor had a thing about humans, according to the old stories; something to do with his retroactive ancestry, apparently". (PROSE: Alien Bodies)

Human

By other accounts, the First Doctor and the Second Doctor were human beings (TV: The Evil of the Daleks, PROSE: Doctor Who and the Daleks, The Monsters from Earth) whose powers of renewal were a function of the TARDIS they piloted. (TV: The Power of the Daleks) The Daleks once believed that it was as a result of his many travels through time that the Doctor had become "more than human", and the Second Doctor himself suggested to the Daleks that he might otherwise be a potential source from whom the Human factor could be sampled, although he also told Jamie that he came from another planet. (TV: The Evil of the Daleks)

When pleading with Edal, the First Doctor referred to himself as a human being, speaking of "human beings, like you and me". (TV: The Savages) He also referred to both himself and Susan Foreman as humans, remarking to Captain Maitland that "[cats] can see better than we humans, because the iris of their eyes dilates at night". (TV: The Sensorites)

One account referred to the Doctor as the greatest human mathematician, whose equations had at long last united space and time fully into the inextricable concept of the Idea of the Living Matter. This had allowed him to construct the TARDIS, a machine which could withstand and travel through Eternity and Infinity in a microsecond. (PROSE: The Equations of Dr Who)

The Third Doctor once visited a parallel universe where he encountered a version of himself who lived on Earth. This Doctor did not recognise the TARDIS nor did he have any knowledge of extraterrestrials or space travel. (COMIC: Who's Who?)

While in a human persona named John Smith, the Seventh Doctor wrote a children's story called The Old Man and the Police Box, about a human scientist from Victorian England who invented a time-travelling police box, established civilisation on the primitive planet Gallifrey, then fled back to Earth after the Gallifreyans become staid and obsessed with law. (PROSE: Human Nature)

Other possibilities

In the post-War universe where the Time Lords never existed, (PROSE: The Adventuress of Henrietta Street) the Doctor was originally a member of the Council of Eight named Soul, who lost his memory following a confrontation with the Eighth Doctor. (PROSE: Sometime Never...)

When the vampire Joanna Harris, having difficulty believing in Time Lords, asked the Eighth Doctor why he hadn't simply introduced himself as "Osiris, or the archangel Gabriel", the Doctor replied with dismay, "You guessed." He then grinned and winked, and James Court, who witnessed the interaction, decided to count this as a joke. (PROSE: Vampire Science) Osiris was king of the Osirians prior to his murder by Sutekh. (AUDIO: Coming to Dust, PROSE: The Sands of Time, et al.)

The Doctor occasionally appeared to his companions as something godlike or monstrous. Bernice Summerfield saw the Seventh Doctor as a transdimensional monster "crammed down into a parody of human flesh". (PROSE: Transit) Mavic Chen believed that the First Doctor's resemblance to "an Earth creature" was merely "a disguise". (TV: The Daleks' Master Plan)

Roz Forrester suggested that he matched the description of Nyarlathotep, (PROSE: The Death of Art) the darkest and greatest of the Old Ones. (PROSE: The Quantum Archangel) When the Eighth Doctor displayed knowledge of the name the Great Old Ones gave to the "Fire of the Last Birth" at the beginning of the universe, Compassion suggested that he might be "a Great Old One on [his] mother’s side". (PROSE: The Taking of Planet 5)

Later developments

The Second Doctor was infused with Androgum genetic material by Dastari, making him 50% Androgum. He began to mutate as it became the dominant factor. He was soon 80% Androgum, and the Sixth Doctor stated that by the time the effect reached him, it would be close to a hundred percent, and that he was already beginning to feel changes. Dastari did not get a chance to stabilise the Second Doctor's cell structure, so his body eventually rejected the transfusion. (TV: The Two Doctors)

The Third Doctor was briefly mutated into a Primord through physical contact with Liz Shaw, who had been exposed to Stahlman's ooze. (AUDIO: Primord)

The Fifth Doctor allowed himself to become a vampire so he could stop Ruath's plan to resurrect Yarven. Following this, he changed himself back to normal. (PROSE: Goth Opera)

The Seventh Doctor, pursued by a family of Aubertides, purchased a device to biologically transform himself into a human named John Smith, complete with a life's worth of false memories. (PROSE: Human Nature) Later, the Tenth Doctor, pursued by the Family of Blood, used a chameleon arch to change himself into a human, again named John Smith. (TV: Human Nature) This same story happened many times, in many ways, to different incarnations of the Doctor. (WC: Shadow of a Doubt)