Many of the 1st and most of the 2nd Doctors' episodes are unfortunately lost. One or two parts still exist of many of these, and there are novelizations of them. I've only listed here the ones that are complete (or restored with animation or narration.)
1st Doctor - William Hartnell.
1) An Unearthly Child
This is the pilot episode, but while part one is a must-see, the rest is a boring story about cavemen and can be skipped.
2) The Daleks
This, along with "The Dalek Invasion of Earth", was remade as a feature film staring Peter Cushing as a human inventor named "Doctor Who." It is officially available for free viewing on YouTube.
* 3) The Keys of Marinus
* 4) The Dalek Invasion of Earth
Like "The Daleks" - this was remade as a feature film. It too used to be available on YouTube, (and is much better than the first movie) but it seems to have been pulled. You can see the trailer here.
2nd Doctor - Patrick Troughton
As I said before, most of his episodes are sadly missing. It should be noted that in interviews, most of the classic Doctors said he was their personal favorite.
5) The Tomb of the Cybermen
6) The Invasion
Although there are missing episodes, the audio for them has been recovered and animated for the DVD (With a "Bad Wolf" thrown in for good measure.)
* 7) The War Games
The first mention that the Doctor is a Timelord, and the first time we see Gallifrey (although it's not called that here.) Last of the black and white episodes.
3rd Doctor - Jon Pertwee
9) Terror of the Autons
Although there were two villainesque Timelords shown earlier, (the Monk in "The Time Meddler" (missing episode) and the War Chief in "The War Games," this is the first appearance of the Master (Roger Delgado)
10) The Time Monster
* 11) The Three Doctors
The 10th anniversary special, with, as the name suggests, includes all three Doctors. Introduces Omega, one of the three founders of the Timelords.
12) Frontier in Space / Planet of the Daleks
Technically two episodes, but one cliffhangers into the other.
13) Invasion of the Dinosaurs
A personal favorite, because it was the first I ever saw at age 9 or so. Wonderfully horrible sets that shake. Hand-puppet dinosaurs badly green-screened over film and stills. Evil eco-terrorists. What more could you ask for?
4th Doctor - Tom Baker ("my" Doctor.)
* 14) Genesis of the Daleks
While not the first Dalek episode, it - as the name suggests - explains how and why the Daleks came to be, and introduces their insane creator, Davros.
15) Pyramids of Mars
16) The Brain of Morbius
17) The Hand of Fear
"Eldrad MUST live!" (R.I.P. Elisabeth Sladen.)
* 18) The Deadly Assassin
A virtual reality world run by a computer called the Matrix that is constantly rebuilt by its users' minds? Sounds like a 100% original idea for a movie to me! if only Keanu Reeves wasn't in kindergarten at the time... (Insert sarcasm where appropriate.) Also, the first and best appearance of the "zombie Master." Introduces Borusa, the Doctor's teacher, who seems to regenerate quite a lot for a Timelord politician (and for some reason, always into old men.) Also introduces Rassilon, the founder of Timelord Society (cameoed by Timothy Dalton in the new series,) along with Omega and that other guy...
19) The Face of Evil
* 20) The Talons of Weng-Chiang.
This episode has been called racist due to the Charlie Chan-esqe casting of it's main villain by an Englishman in Asian makeup and it's stereotypical Chinese characters, but it's still a great story.
21) The Key of Time
This story arc consists of six episodes: The Ribos Operation, The Pirate Planet, The Stones of Blood, The Androids of Tara, The Power of Kroll, and The Armageddon Factor.
* 23) City of Death
Co-written under a pen name by Douglas Adams and guest-starring Julian (he who chose... poorly) Glover. Parts of this story, along with "Shada," were incorporated into the novel Dirk Gentley's Holistic Detective Agency. Filmed on location in Paris.
Written by Douglas Adams, incomplete and never aired due to a BBC strike. It was released on video in the 90s, with Tom Baker narrating the missing parts (which, of course, are the most expensive action shots.) Was re-made as a Big Finish Audio Adventure with the 8th Doctor starring in Tom Baker's place. See "City of Death."
25) Full Circle
*I HATE Adric, but it is a good story.
Tom's regeneration, and first full story of Anthony Ainley as the 3rd (or rather, 14th?) Master.
5th Doctor - Peter Davison
28) Arc of Infinity
30) Mawdryn Undead
A nice return episode for the Brigadier. Re-introduces the Black Guardian (See The Key of Time story arc,) but ruins the impact with a very cheesy outfit and a sense of powerlessness.
A good story. No, seriously - it's not just here because Nyssa strips down to her underwear for no good reason...
* 32) The Five Doctors
20th anniversary special, with all five Doctors. Well... not quite. The late William Hartnell was re-cast with a look alike, and Tom Baker had a bug up his ass, so he refused to do it, (he appears using clips from the unaired episode "Shada" and is "caught in a time bubble") Although the story is interesting, it suffers from too-many-character-itis. As a result, Susan and Turlough are superfluous and spend most of the episode locked in the TARDIS.
Cashing in on the upcoming Star Wars hype, this was unfortunately released as a "Special edition" in the 90s. The 80's special effects were updated, but just like the Star Wars special editions, the new CGI looks out of place with the rest of the episode, and look just as dated fifteen years later as the original effects did when the SE was produced. I'm also not fond of the attempt to digitally deepen Rassilon's voice. On the other hand, the special edition does contain extended shots and some deleted footage, so it's a give and take.
* 33) The Caves of Andriozani
Peter Davison's regeneration episode.
6th Doctor - Colin Baker (no relation to Tom.)
I initially wasn't a fan of the 6th Doctor, but once the BBC realized that "complete unlikable asshole" wasn't a good character choice and they toned him down a bit (if not his clownish outfit) his episodes became much better. He makes such a good Doctor, in fact, that he is my favorite one of the Big Finish Audio Dramas - but that's a list for another post.
* 34) The Two Doctors
Patrick Troughton's last appearance as the Doctor, along with Jamie. The fact that they are both about twenty years older is ignored. (See the Season 6B theory.) Filmed in Spain.
* 35) Revelation of the Daleks
Has one of my favorite characters - Orcini (for those of you who have known me since the AOL days,) and co-stars Eleanor Bron. My favorite Dalek story of the classic series.
36) The Trial of a Timelord.
Sometimes called a 14 part story, sometimes broken into four smaller stories (Three four episodes long, and one two eps. long) It introduces the very important (though never heard from again) character of The Valeyard.
7th Doctor - Sylvester McCoy.
It should be noted that at first the 7th Doctor is portrayed as silly and fun, but later on is transformed into a dark, brooding, manipulative figure. This is aspect much more explored in the Virgin New Adventures novels after the show's cancellation.
37) Delta and the Bannermen
38) Remembrance of the Daleks
* 39) Silver Nemesis
The 25th anniversary story - hence the name. Gives a little hint about that other guy...
40) Ghost Light
The last episode of the classic series.
8th Doctor - Paul McGann.
Unfortunately, the 8th Doctor had just one episode. It's a TV movie called "Doctor Who" or sometimes, "The Enemy Within." It was the pilot for an American produced continuation of the series in the 90s that was never picked up. Paul McGann is a fine Doctor, Daphne Ashbrook makes a lovely companion. Eric Roberts as the Master... meh. (I had a beard - why didn't they call me?) Good production values, but basically a very long regeneration scene and a lot of chasing. Notable for a blasphemous attempt to ret-con that the Doctor is half human. Paul McGann did an awesome job picking up the character of the Doctor on the Big Finish Audio Adventure series, and in BBC radio dramas (which were also produced by Big Finish.)
Notable external media:
Dimensions in Time
A mash-up for the Children in Need charity between Doctor Who and EastEnders for the 30th anniversary of Dr. Who (1993.) Utter, horrible crap, but it's fun for nostalgia's sake. The 3rd - 7th Doctors and most of his companions run around Albert Square and interact with EastEnders characters. The last appearance of Jon Pertwee as the Doctor.
* Doctor Who and the Curse of Fatal Death
Rowan Aktinson and other celebrities star as the Doctor in a spoof for charity. Penned by Stephen Moffatt, who currently helms the new series. Jonathan Price steals the show as the Master.
The last of the 7th Doctor novels, and the final explanation of... that other guy. Although the new series has refuted it in some ways and some people question whether it is canon, this is so far, the only official origin story of the Doctor. It was supposed to be part of the original series ("Remembrance of the Daleks" and "Silver Nemesis" build up to it,) but Jon Nathan Turner got cold feet, and the series was cancelled before it could be made into an episode. Definitely a must-read.
My list of Big Finish Audio Adventures deserves its own post, but this was Doctor Who's 40th anniversary story. Again, Tom was a bastard and didn't join in, but it reunites the 5th, 6th, 7th, and 8th Doctors. Notable for having actors who played the Doctors and companions from the original series playing other characters rather than trying to clumsily shoe-horn why everyone who ever knew the Doctor is in one place.
* Unbound (series)
A Big Finish Spinoff series, each story asking "What if?" (For example: What if the Doctor never left Gallifrey? What if the Valeyard had won? What if The Doctor was a woman?) Most notable actors playing the Doctor in these are Sir Derek Jacobi (later to cameo as the Master in the new series) and David Warner (who I would love to see as the next Doctor, if the BBC could please just get over the squee factor.)
Anything I forgot?