Part and parcel of why fans find a change of Doctors exciting and/or terrifying, I think, is that it's usually more than a change of actors. It's a change of approach, and usually a change of producer and/or script editor causing the program to bend and shift according to the vision of all these new people. It's why you'll also find "favorites" discussions in Who fandom divided into "favorite Doctor" and "favorite era", the two questions not always yielding the same result (when asked I might be a Troughton or McCoy man, but still prefer the 4th Doctor's early years to the rest of the program, for example).
Each era has a unique feel that goes beyond its Doctor, but usually informs his performance (though I'm sure the reverse is also true), which builds on the previous era as to not alienate the viewership. And yet, they often do, before they charm your pants off all over again. Maybe looking at the past will allow us the look at the future more open-mindedly?
1st Doctor - The Beginning (1963-1966)
Though designed as an educational adventure program, the Daleks' first appearance in the very second story laid the groundwork for most of what was to come. This is the golden age (some might say the only age, and they would be correct) of the pure historical as the Doctor and his companions visited various periods of history, from the stone age to the siege of Troy to the French Revolution. The Doctor was a sort of anti-hero, largely leaving the heroics to the younger cast members, but the format remained quite elastic. They were making it up as they went along, so you have a lot of variety both in length and subject matter - historical strage plays (The Aztecs), monster movie madness (The Daleks), surrealism (The Edge of Destruction), comedy (The Romans), they all jockey for attention and in a sense gave Doctor Who license to go anywhere it wanted to in later years.
A good example of the era: The Aztecs
2nd Doctor - Start Running (1966-1969)
Admitting to itself that the monster stories were a hell of a lot more popular than the historicals, Who then started making them almost exclusively. With a younger sprightlier Doctor, they had the chance to do a lot more running away from the creatures that took over the show. The Daleks appeared a couple times before being temporarily taken away by their creator and rights-holder, Terry Nation, so the Cybermen became the Big Recurring Bad. The era also introduced the Ice Warriors and Yeti, which both made two appearances.
Example: Tomb of the Cybermen
3rd Doctor - The UNIT Era (1970-1975)
Then came producer Barry Letts and script editor Terrance Dicks, oh and colour tv. Stranding the Doctor on Earth in the present day for the largest portion of his time as Jon Pertwee meant better production values, more explosions, more stunts, at the expense of production design. Obviously taking its cue from James Bond's success, this is the superspy era. The Doctor has never been so hands on, driving fast cars and karate chopping his way to victory. He had a Bloefeld-like enemy in the Master and companions were distilled into a single assistant, sparking the Bond-like term "Who girls". Though some contend that having him work with UNIT, which is, after all, a military organization, went against the character's anti-establishment vibe, but he definitely worked outside their hierarchy and butted heads with bureaucrats and military men on a regular basis. Though it may seem like a total departure from the 2nd Doctor's era, UNIT did get its start in 1968 (The Invasion), and of course, Doctor Who would never shake its "monster of the week" format.
Example: The Sea Devils
4th Doctor - The Hinchcliffe-Holmes Era (1975-1977)
Leaving UNIT behind, producer Philip Hinchcliffe and script editor Robert Holmes (universally acknowledged as the best writer of Classic Who) reformated the series one again. Under their regime, Doctor Who got more atmospheric and took its cues from the Hammer Horror films of the time. The Doctor and his assistant (a format mostly retained for the rest of the 70s despite characters like Harry and K-9 cropping up) met mummies, disembodied hands, cultists, Frankenstein's monster, the Yellow Menace, body snatchers, etc. all with a science fiction twist. Darker and more violent, it was a natural progression from the explosive action of the previous era.
Example: The Talons of Weng-Chiang
4th Doctor - Hitchhiker's Guide to Douglas Adams (1977-1981)
Tom Baker remained in the role long enough to span two eras, proof that the actor isn't the only element to watch out for. A number of trends are key here. First, the many complaints leveled at the show by "morality watchdogs" like Mary Whitehouse pushed new producer Graham Williams to lighten the mood and brighten the lights. Cue kiddie fodder like K-9, set a lot of stories on white space stations, and get Hitchhiker's Guide mastermind Douglas Adams as your new script editor. Much of this may have been inspired by Tom Baker's own pregressively more cheeky and self-indulgent performance as the Doctor. A "funny" Doctor begat a more comical program, whose greatest weakness was lacking tension. What is there to be scared of when you can laugh everything off as much as the Doctor can? The final year saw uber producer John Nathan-Turner (JNT) take over the show, emphasizing gloss over substance, but Tom Baker had become such an icon by then that one can scarcely talk of a change in eras.
Example: City of Death
5th and 6th Doctors - Glitz and Glam (1982-1986)
Moving from its weekly Saturday night slot to a twice weekly mid-week presentation made Peter Davison's stint as the Doctor something more akin to soaps, an impression reinforced by the addition of three companions in the increasingly crowded TARDIS. JNT's focus wasn't on story, but rather on look and publicity, which isn't a good thing, though it was probably why the show was kept on the air through the rest of the 80s despite the shoestring budget and diminishing returns. Hallmarks of the JNT period include few-to-no costume changes (the action figure principle), highly publicized companion changes and returns for classic enemies, location shoots to exotic countries (with no real scripted reason), sometimes miscast high profile guest-stars, and trying to start hairstyle trends (poor Janet Fielding). It's easy to criticize, but perhaps the high profile of the show during Tom Baker's turn made the environment ripe for more media attention, and perhaps JNT was desperately trying to keep the show in the public eye even as interest for it waned in the wake of Baker's departure.
I kind of feel bad for script editor Eric Saward actually. From every interview I've seen, this soft-spoken man seems to have been constantly put upon to incorporate JNT's shopping lists of "cool" media stunts, even in the face of logic. Perhaps that's why his own stories were "mercenary" stories with more attention given to guest-starring characters than the regulars (The Visitation, Revelation of the Daleks). In the 6th Doctor, we have the purest JNT vision. The Doctor has a terrible action figure costume, the assistant is American so that the program can be sold to the Yanks (despite Nicola Bryant struggling with the accent), and the sex and violence are on the rise.
Examples: Resurrection of the Daleks, Revelation of the Daleks
7th Doctor - The Cartmel Plan (1987-1989)
But I'm still not keen on lumping all the 80s in some kind of macro-JNT era, because while their production style is the same, the shows scan differently. JNT's lack of interest in story simply means that the era is much more dictated by the script editor. And under Andrew Cartmel, Doctor Who became a very different animal. Yes, the first few stories suffered from being "pants", as the Brits call it, but as Cartmel got into his groove, the Doctor was turned into a temporal mastermind with a definite dark side. He manipulated situations, villains, even companions. Writers were attempting much more intricate stories to fit this vision, some of which border on the incomprehensible to some, like the textured and allusive Ghost Light. Regardless of the upswing in quality, the Doctor's days were numbered since Colin Baker's time and the BBC finally pulled the plug. Cartmel's vision of the Doctor would continue his journeys in the New Adventures series of novels, which would extend the complexity and darkness of the character and stories much further, and provide a forum for writers who would go on to work on New Who, like Mark Gatiss, Paul Cornell and... Russell T Davies.
Example: The Curse of Fenric
8th Doctor - Aborted American Era (1996)
Well, that didn't work out. What WOULD this era have been like had Fox picked up the series instead of Space Above and Beyond though? American companions, actors and locations, most certainly. The innovation, of course, is the Doctor as a romantic lead, which had never been attempted before (outside a few flirtatious moments in the 1st Doctor's era). And it's something that would carry over in the new series. You can follow Paul McGann's Doctor in a long line of novels and audios, which are a mix of New Adventures and the TV Movie (amnesia is a recurring theme, for example).
Example: The TV Movie
9th and 10th Doctors - Rose (2005-2009)
It's not that Christopher Eccleston stayed too briefly to have his own era, but rather that New Who up til now has been Russell T Davies' story above all. Starting with "Rose", it's a story that only finds its conclusion in Journey's End, 4 years and 2 Doctors later. RTD reinvents the program for the 2000s as a slick, effects-heavy, fast-paced, emotionally charged show. Whether Eccleston or Tennant, the Doctor is a romantic lead in RTD's mind, which I don't think is due to change, as new producer Steven Moffatt's episodes over the last 4 years have been heavy on romance. The same can be said of the epic nature of the stories, painting the Time Lord as a Time God (so echoes of both the 7th and 8th Doctors). Throughout it all, it remains recognizably Doctor Who.
Examples: The Parting of the Ways, Doomsday
11th Doctor - Moffatt's Reign (2010-?)
Obviously an extention of New Who to date, with one of its best writers turned producer, the show will no doubt be recognizable, but I hope sufficiently different too. Because that's what Doctor Who is all about, isn't it? Change and resistance to conformity.
It's looking like I'll be doing these pieces all week, so I think I'll be taking questions. Anything Who - themes, trivia, opinions, theories, anything you like. Those not answered over the course of pre-planned essays will be answered in Friday's post. Anything you'd like to know or discuss?