First Doctor: the renewal
The funny thing is, at no point in the change from Hartnell to Troughton does anyone ever call it "regeneration". As a matter of fact, the sorely weakened Hartnell doesn't say much anything about what's going on at all, leaving a stunned Ben and Polly to spend much of the first episode of "Power of the Daleks" trying to figure out what's going on and whether this short new person is who he says he is. We learn a few things about the process; the Doctor calls it "renewal", says it couldn't happen without the TARDIS, and seems reasonably pleased with his appearance and new body. David Whitaker has a delightful line in mixing hard science fiction with a metaphorical, philosophical tone (the alchemical "mirrors and static electricity" time machine in "Evil of the Daleks" is probably the single clearest example of same). At this early stage in development, such mystery still suits the series well.
(Whatever the change is, the production team don't think it's solely a matter of biology right now; the Doctor's clothes alter along with him, though his ring doesn't. Compare and contrast with the way that a specially large pair of cricket whites was made for Colin Baker so that he could appear to still be dressed in the same outfit as his predecessor.)
Second Doctor: the time the Time Lords did it.
And the Doctor's not happy about it. Once they insist that they're going to exile him to Earth, he starts quibbling about his appearance being known there. Exactly who he's worried about running into isn't quite clear; perhaps he wasn't as keen on running into Lethbridge-Stewart as all that? The Time Lords take advantage of this (or, quite possibly, had already decided) to force a new body on him, much to his chagrin (the TV Comics ignored this and had an exiled Second Doctor living it up on 60s Earth for quite a while before the Pertwee stories started, the first iteration of what fans would go on to call "Season 6B"). It certainly takes the new Doctor longer to recover than last time, spending an entire episode in hospital and escaping in a wheelchair because he can't run yet. The first thing he does in "Spearhead from Space" is fall out of the TARDIS, so perhaps he left before the regeneration fully "set". Initially he's not best pleased with the enforced new look, but Pertwee gets used to it soon enough.
Third Doctor: the time that it's actually called "regeneration".
Because Barry Letts had recently taken up Buddhism and was excited about applying his new toolkit of concepts to the show, including the concept of becoming a new man more literally than is usually meant by the term. Practicing Buddhists weren't unilaterally thrilled about the representation of their beliefs. Unlike most regeneration stories when the cause appears to be more or less arbitrary (the Doctor doesn't actually *need* to fall off a radio tower for the rest of "Logopolis" to work as a story), it's made thematically clear that it's this Doctor's time to go; Pertwee telling Sarah Jane that he had to do it because of his lust for knowledge is one of the most heart-sickening moments in all fifty years worth of the show.
There's an implication that the Doctor's regeneration has actually got stuck somewhere along the line, as it seems to require the intervention of K’anpo to kick-start his alteration into his new self. (Paul Cornell's Love and War takes this idea to extremes and suggests that the Doctor spent a terrifyingly long time stuck in the vortex on the way from Metebelis Three to Earth, on the order of ten years or so. Which rather begs the question of how the TARDIS could possibly take that long to get from A to B, especially since the Doctor says she's the one who brought him home, but that's a matter for the spin-offs to work out.)
Tom Baker seems to be having a little trouble settling in in "Robot," but so much of what happens there will turn out to be integral to his character (the general disinterest in UNIT, the rapid-fire nonsensical jokes), it's hard to say what's post-regenerative confusion and what simply amuses the new Doctor. He muses on his appearance, but the part everyone remembers is the UNIT scene in which a world-weary Brigadier stares incredulously at the Doctor playing dress-up.
Fourth Doctor: the radio tower time
In which we learn that, contrary to what was implied in "City of Death", Time Lords don't fly. Not does it help that he's already regenerated before he gets to the TARDIS. There's a hint that Peter Davison's regeneration goes badly because of sympathetic pain from a quarter of the universe being blown up at the same time, so his companions build him a coffin to help him recover. Of course, taking your first trip outside the TARDIS in the Master's personal mathematical fantasy land couldn't help much, and at least he's on his feet by episode four.
Fifth Doctor: the painfully bonkers time
Odd as it is to die from poisonous bat guano (in addition to his finesse as a writer, Robert Holmes liked his rude jokes), by the time anyone's finished watching "The Caves of Androzani" they probably have quite a lot of good will for what comes next. Unfortunately, Eric Saward had a Clever Plan to start off with a nasty Doctor and turn him into a nice one by degrees. This is a storyline that would proceed to go on for entirely too long in the next season, and is played up in "The Twin Dilemma" to such an extent that even if the rest of the story had been fine, the Doctor's character would still have been unspeakable. Colin Baker tries fervently to do something watchable with a character who witters on self-indulgently with a complete lack of regard for his companion, but it's his first go and the material is unrewarding. Nor is it very clear why the regeneration is working out so badly this time; surely the whole point of a regeneration within the TARDIS is that it should sort out all your biology properly. Is raw Spectrox really that poisonous?
(The novelisation works slightly better as a story because Saward's dark and sardonic sense of humour translates better to a prose style than it does to the screen, and you don't have to look at the eye-watering colours; it reads like the Douglas Adams who wrote Mostly Harmless, but with more sadism and general unpleasantness.)
Sixth Doctor: the cheerfully camp time
We have no idea what even caused it, at least on screen; it looks like the Sixth Doctor simply bumped his head and died. Gary Russell, who's quite a fanboy of the Sixth Doctor, wrote a BBC novel "Spiral Shock" that tried to make this slightly more meaningful; the plot entails a dimensional breakdown and the Doctor sacrificing himself to save the whole of existence (actually, he dies from an overflow of temporal energy. The book was published mere weeks after "The Parting of the Ways."). And then Chris McKeon used some notes and plans of the late great Craig Hinton to write "Time's Champion". This is an entirely different story about the Doctor saving the whole of existence from a dimensional breakdown just before he regenerates, which ends by setting up the events of "Spiral Shock" (yeah, this one requires an A-level in Doctor Who continuity to understand; it's a novel written in honour of the man who invented the word "fanwank"). All of which is probably still more cheerful for Colin Baker fans than the Virgin New Adventure version that has the Seventh Doctor more-or-less killing off his past self. It's complicated, see "Love and War" again if you want further details.
The Doctor regenerates in the TARDIS before the Rani makes off with him, which perhaps accounts for his generally stable-if-grumpy new self; it takes most of the story for him to settle down but he's not nearly as irritating about the transition as he was last time. By now the whole body dysmorphia and general state of confusion is a series tradition, but at least the Rani running interference means there's a story reason for him to be uncertain for much of the screen time.
(And his clothes didn't regenerate this time either, which means Sylvester McCoy spends most of an episode running about in the same multi-coloured
Seventh Doctor: the time that's what the whole story is about
So the mysterious and manipulative Seventh Doctor, who by this time has had fifty novels worth of character growth and new companions, dies in a complete accident involving his having two hearts and the state of the American health-care system (the last few Virgin New Adventures tried hard to justify this, with Kate Orman "The Room With No Doors" getting in some excellent meditation on the nature of change). Paul McGann is left to regenerate in a morgue and comes out of it physically fine, but with a case of amnesia that would inspire an enormous number of BBC novels; not being in the TARDIS might account for some of the trouble here. A lot of people have tried to force the "half-human" line into applying only to this Doctor in particular via some complicated regeneration mechanism or other, although that's clearly not the intent of the script and it's much easier to presume that the confused Time Lord is just talking cobblers.
Eighth Doctor: the time nobody knows about
And no one's likely to any time soon; the BBC isn't hugely keen on storylines that require TV license payers to pay extra money for extraneous content, which scuppered DWM's plans to tell the regeneration story in the comics (this despite the various computer game tie-ins that are required to even start making any sense of "Asylum of the Daleks'). You can still see the general shape of the intended story in "The Flood," the last Eighth Doctor comic the magazine ran before switching to Ninth Doctor stories. Davies decided it'd be best to start fresh and not worry about continuity at the time, so unless there's a movie version involving McGann (improbable at best), we're not going to get an answer on this for a long time to come. Maybe in twenty years when Big Finish have the rights to Ninth Doctor audios?
Ninth Doctor: the time everybody knows about
"Bad Wolf/The Parting of the Ways" had its flaws, and plenty of the storylines it brought up were never resolved satisfactorily, but there's no denying that Russell T Davies created a gorgeously apt send-off for Christopher Eccleston. It built on what the rest of the season had established, it allowed Rose to have her most heroic moment in all her appearances, and the regeneration was both dramatically appropriate and affirming. Probably still the best season-ender in all of New Who.
"The Christmas Invasion" has a good mix of post-regenerative tropes and is festive enough to get away with all of them; there's enough story going on for the Doctor to enter it at a satisfying point, and the "choosing clothes" business tops it all off nicely. On the other hand, it's the story that established the "fifteen hours, free limbs or your money back" business, and that would be...unfortunate, later on. Also the first hint that regeneration is aided by stuffing your face at the earliest available opportunity (see also: John Simm and roast chickens, fish fingers and custard).
Tenth Doctor: the very, very long time
Twenty minutes worth of screen time spent in the process of regeneration; by the time the Ood have come along, it looks like David Tennant is about to explode five hundred thousand rads worth of radiation right into the Powell Estate (at least it adds a bit of last-minute tension the first time). No, it's a very appropriate send-off for Davies, Tennant, and the teams that made the Doctor Who revival so successful, even if it is a little self-indulgent, and the tributes to the spin-off shows is rather nice. Due to the way that Steven Moffat writes scripts, we still don't know if there was anything going on in "The Eleventh Hour" that we don't know about yet, but it seems fairly safe to say that the Doctor's various temporal errors are meant to be written off as post-temporal confusion; once Matt Smith has calmed down enough to sort out his TARDIS steering, he's stable enough.
So there you go; ten regenerations in two thousand words. How Matt Smith's exit will go remains to be seen...