Despite canon not really being a concept in Doctor Who, there's generally broad agreement over who "counts" as a Companion. The areas of contention are usually around the short-term companions, and how to distinguish between them and guest actors. The line isn't always so clear.
The closest thing to an "official" word is this list of classic Who companions posted on the BBC website, or this list of all companions. In the interests of archiving (with asterisks next to the contentious ones):
First Doctor
- Susan Foreman
- Ian Chesterton
- Barbara Wright
- Vicki
- Steven Taylor
- Katarina
- Sara Kingdom*
- Dodo Chaplet
- Polly
- Ben Jackson
Second Doctor
- Jamie McCrimmon
- Victoria Waterfield
- Zoe Heriot
Third Doctor
- Brigadier Sir Alistair Gordon Lethbridge-Stewart
- Liz Shaw
- Jo Grant
- Sergeant Benton*
- Captain Mike Yates*
- Sarah Jane Smith
Fourth Doctor
- Harry Sullivan
- Leela
- K-9
- Romana (two regenerations)
- Adric
- Nyssa
- Teagan Jovanka
Fifth Doctor
- Vislor Turlough
- Kamelion*
- Peri Brown
Sixth Doctor
Seventh Doctor
Eighth Doctor
Ninth Doctor
- Rose Tyler
- Mickey Smith
- Jack Harkness
Tenth Doctor
- Martha Jones
- Donna Noble
- Wilfred Mott*
Eleventh Doctor
- Amy Pond
- Rory Williams
- River Song
- Clara Oswald
Twelfth Doctor
Controversies
I asterisked a few names up there; why did I do that?
- Sara Kingdom is not always considered a companion; she was very briefly connected to the Doctor, following him purposefully (rather than getting trapped on the TARDIS, which is how he normally meets girls), with a mission to complete, and then dies. Narratively, she had more in common with a guest actor than with a companion
- Sergeant Benton never travelled on the TARDIS, although he did enter it in The Three Doctors, which is more than Captain Yates can say. He's mainly included because the Third Doctor's run didn't involve much TARDIS travel, and Benton was part of the UNIT ensemble cast
- Mike Yates see above
- Although Kamelion fits most reasonable definitions of "companionship", because of technical difficulties with the puppet he only appeared in two episodes: his introduction, and his destruction (and one more, through the magic of archive footage). Though he got somewhat fairer treatment in the novels, you could also make a plausible argument for leaving him off
- Wilfred Mott was an Earth-based recurring character in series 4, but he's not really considered a companion until the two-part "The End of Time"; in that regard he bears more in common with Christmas special one-off "companions", so it feels slightly unfair to count him without counting them
Omissions
In the Night of the Doctor minisode, the Eighth Doctor rattles off a list of names before declaring them "companions" (emphasis mine):
Doctor: Get out. Get out! All of you. Will it hurt?
Ohila: Yes.
Doctor: Good. Charley, C'Rizz, Lucie, Tamsin, Molly. Friends, companions I've known, I salute you. And Cass, I apologise. Physician, heal thyself.
Doctor Who Minisode "The Night of the Doctor"
These characters (except for Cass) never appeared on-screen; they were introduced in the Big Finish audios, which is probably why they weren't in the companion list.
Whether or not you consider Cass a companion is another open question.
Adam Mitchell, from "Dalek" and "The Long Game"1 is also omitted. Whether or not you consider him a companion is, again, up to you
Handles, the Cyberman head the Doctor was carrying around in "The Time of the Doctor"
What about the TARDIS herself? She's been established as (at least) semi-aware since "The Edge of Destruction" in 1964, and has appeared in just about every episode
The BBC list is notable for not including one-off characters from, among other things, new-Who Christmas specials. It's mainly up to you if you consider these characters to be companions. They serve the same narrative function, but they're also just guest actors
- Astrid Perth ("Voyage of the Damned")
- Jackson Lake ("The Next Doctor")
- Christina de Souza ("Planet of the Dead")
- Adelaide Brooke ("The Waters of Mars")
- The Arwells ("The Doctor, the Widow and the Wardrobe")
- Kazran Sardick and Abigail ("A Christmas Carol")
- Craig Owens ("The Lodger" and "Closing Time")
As I mentioned above, Wilf kind of deserves to be on this list; although he has several appearances, he's only ever on the TARDIS in a single (two-part) episode. Though I haven't gone through and checked, I believe he spent less time on the TARDIS than Jackie Tyler did, so make of that what you will.
Definitions
The simplest definition of a companion is this:
Companion. n. Someone who travels with the Doctor on the TARDIS
But it's not a perfect definition:
- It includes a broad class of people who we wouldn't normally consider companions; Rigsy, for example
- It excludes some people who we normally would consider companions, like Liz Shaw and Grace Holloway
- It cuts right down the middle on one-off characters. Adelaide Brooke and Kazran Sardick are in; Astrid Perth and Christina de Souza are out.
Okay, let's try a different one:
Companion. n. A character, other than the Doctor, played by an actor whose name appears in the opening credits.
This definition works well enough for the new series (though it excludes Adam, and possibly others), which has had a pretty consistent opening credit scene ("Sleep No More" notwithstanding), but less effective for the old series; the character credits for the old series normally went after the episode, with only the writing credit appearing at the start.
Maybe you could tweak it slightly:
Companion. n. A character, other than the Doctor, played by an actor who gets independent billing in the credits.
A little better, but I picked a random episode from my library and checked the credits; look who this definition includes:
We can't have some upstart Sontaran trying to muscle in on companionship2. That won't work either.
You could make the argument that the companion is meant to be an audience surrogate character, the person who gets technobabble explained to them and who enforces a human morality for audiences to connect with. That's fair enough, I guess, although I'm not sure I'd agree that's a good definition. For the sake of argument, though, that largely excludes Captain Jack, Nardole, and (arguably) River and K-9.
This is starting to give me a headache.
As with most things in Doctor Who canon, a companion is whatever you say it is
1 Nod to Josh for reminding me; Adam was so unremarkable that I completely forgot about him
2 Oh, hello Strax. I didn't see you there
New Who<->Sarah Janeis a thinko (like a typo, but in thought process), since Sarah Jane was an old Who companion who only appeared for one episode in the new Who.