Hartnell: Assuming the newbie in question gets the premise of the show (guy wanders around time and space in astounding vehicle disguised as oldskool phone box with various travelling companions defeating the forces of Evil TM, it's not a terribly complicated concept), I'd skip the introductory episodes (An Unearthly Child the episode rocks, but the guff with the cavemen drags a fair bit) and go straight for The Dalek Invasion of Earth, which is a wonderful example of a production team less than a year into the job deciding to push the envelope and see how far they can go. The sheer chutzpah of thinking they can get away with Daleks invading Earth on such a massive scale - and, for the most part, it works, aided by some wonderfully eerie shots of deserted London streets. It's a good, solid, six episode adventure, with not a wasted moment.
Troughton: I'm going to cheat slightly, and go for something that doesn't exist in visual form - The Power of the Daleks, which is quite simply my favourite Troughton story. It's also a good way of introducing newbie to the notion of regeneration.
Pertwee - You'd have to briefly explain the whole notion of the Doctor having been exiled to Earth, but I'd recommend Carnival of Monsters, which works fairly well as a standalone, but also is wonderful, cheerful fun. And Katy Manning is in it.
Baker, Tom: I think I'm going to cheat again, and nominate two - which is fair enough, as his time as the Doctor was twice as long as many others. Both stories that work well as standalones, and in their own ways capture the spirit of the programme whilst also being fairly adventurous with what you can do within the format - Talons of Weng-Chiang and The City of Death.
Davison - Ooh. Difficult. It's tempting to go for Kinda, but that's maybe a tad too complex; for Snakedance, but that probably requires watching Kinda first; or for The Caves of Androzani, but I think that's a special treat and not for newbies. Or Terminus, which I adore despite most of fandom having decided that it's pants. Earthshock is great, but I'll actually go for something fairly low key, The Visitation, which is just A Good Example of The Sort of Thing Doctor Who Did in the 1980s, and darned enjoyable. And it's got Michael Robbins being awesome.
Baker, Colin - I can't be objective about this, I'm going to recommend The Two Doctors. I know it's not the greatest story in the world, but it's got two of my favourite Doctors being awesome. I don't care.
McCoy - Again, I'm going to buck the trend here and recommend something a bit left field - Delta and the Bannermen, just because it's a bit lighthearted and frothy, doesn't require a masters degree in the programme's history, has Bonnie at her best and Sylvester not being too embarrassing, and is worth 75 minutes of anybody's time.
McGann - Ooh, tough one. Well, he was in a film called Withnail and I, that's worth checking out...
Eccleston - Can't be bothered bucking any trends on this one - just go straight for The Empty Child and The Doctor Dances.
Tennant - I'm gonna go with Blink, on the basis that until he takes over as showrunner anything by Steven Moffat rocks. Girl in the Fireplace is actually a better episode (better than just about anything else, IMO), but again, that should be a special treat and not something to be taken lightly. Blink is excellent, in turns scary and witty, and a good example of a Doctor-lite episode. It's also got the absolutely ace Carey Mulligan.
Smith - Bucking the trend here. Other folks have gone for The Eleventh Hour, and I agree that's excellent, but I just want to go for something else. And of course with Smith's two seasons, once you get about half way through the first one you get mired up in arc plots, so I'm going to go with Vampires of Venice, which is another of those good, enjoyable, fun ones, and it's got Arthur Darvill being a swashbuckling hero. Which is worth seeing.