What species is The Doctor?

Actually this is slightly harder to answer than you might think. Throughout the show (and reboot series) he's been described as being either a Timelord or, latterly, a Gallifreyan.

CHRISTINA: You look human.

DOCTOR: You look Timelord. Anyway.

_{^{Planet of the Dead}}

and

MONARCH: [You are] An Earthling.

DOCTOR: A Gallifreyan. Tegan is an Earthling, Adric is an Alzarian, and Nyssa there is from Traken.

_{^{Four to Doomsday}}

However, in the 1996 Doctor Who film, the Doctor was repeatedly confirmed to be half-human(!).

MASTER: Fascinating. See that? That's the retinal structure of the human eye. The Doctor is half human! No wonder.

and

WAGG: Grace says you have a big secret. What is it?

DOCTOR: I'm half human. On my mother's side.

This fact could be easily dismissed as non-canon if it wasn't then confirmed in at least one other property; The Infinity Doctors

The Doctor knew exactly who he was, who he’d always been. He was a Time Lord, from the Noble House of Lungbarrow on the planet Gallifrey. He had been born of the Loom, son of the greatest explorer of his age and a human woman, Annalise… no… his mother’s name had been Penelope. He knew his father’s name, at least: his father’s name wasn’t Ulysses

---

Who Showrunner Russell T. Davies apparently wanted to retcon this (questionable) heritage out of existence, but ultimately chose not to do so out of respect for the film.

“I don’t like the half human thing. He certainly isn’t half human, but it’s less interesting to say it simply doesn’t count. I always wanted to put in a line where someone says to the Doctor, ‘Are you human?’ And the Doctor says, ‘No, but I was once in 1999. It was a 24-hour bunk.’ Part of the reason I never put that in was it was a bit too self-referential but also I thought, ‘I’m spoiling the TV-movie if I do that.'”

_{^{Doctor Who: Half Human or All Time Lord?}}

Unknown.

The Doctor is NOT Gallifreyan (all other answers are wrong (reads: outdated)).

In the recent episode The Timeless Children, we learned that

The Doctor also didn't know this. That's possibly because either Gallifreyans hid this fact from him/her or the information got lost when his/her memories were wiped (it was also shown in the episode that his memories were wiped countless times).