Yes, spoilers. Super super geeky content follows.I'm still thinking about the end of David Tennant's tenancy as Doctor Who in the two-part End of Time episodes.
We don't know much about the Doctor's past, although the Master talks about he and the Doctor playing in the fields of the Master's family estates. So perhaps there were parents (although other parts of the canon make this more complicated).
It appears from clues in the episodes that a time lord who appears on cue to Wilfred multiple times, and as one of two dissenters in the vote on destroying the universe at the end, is the doctor's mother. (It's totally unclear how she shows up so many times when she is in disgrace and was already in the Time War's time lock.)
My original theory, before the very end of End of Time, was that Wilfred was a time lord who had passed through the Chameleon Arch, and was unaware of his own identity, just like the Master was in his Professor Yana garb, and the Doctor as John Smith (also a professor! or don, at least).
I figured that the doctor's mother (we figured that out later) appearing was actually a manifestation of his hidden nature, and that calling him Old Soldier was a kind of code word that was supposed to awaken his interest in some object, like a watch.
Instead, Wilfred is just who he was: an honest, sometimes terrified, perfectly decent human being. The Doctor is undone not by the Master or some race of super-intelligent cyborgs. Rather, he gives up his incarnation because he must do the right thing in the face of someone doing the right thing.
In earlier episodes, the Doctor is always reaching out a hand to even the worst of his enemies: he is always trying to save the Master. He said to Davros that he tried to save him as he flew into the jaws of the Nightmare Child (something that we will probably never know what was meant by); he tries to save Davros after the Doctor's duplicate kills all the Daleks.
So how much worse to have a good man, by his own goodness, require saving?
The actions of the Doctor's double in the Dalek Reality Bomb episodes is finally explained in part in the End of Time. The double kills all the Daleks in a single act with little forethought. Donna tries to stop him, saying, shouldn't we wait for the Doctor? And the double says, "I am the Doctor." Which is true. He is as the Doctor was when he ended the Time War.
The Doctor later shunts his double to be Rose's partner in a parallel world, and he explains that he was born in war and fire, and that his nature is equal to the same. The folks who monitor Doctor Who continuity posit that the Ninth Doctor (Christopher Eccleston) was regenerated during the Time War, so born in the same spirit. (The 10th Doctor, David Tennant, says as much.)
The Doctor mourns the end of the time lords, but he also knew he had no choice. Had he not destroyed his own people and the Daleks in the Time War, then the Time Lords would have destroyed time (becoming somehow pure consciousness in the absence of time and space). The Dalek leader, Davros, having been plucked out of the Time War, tries to destroy all matter--a mimicry of the Time Lords, like all the Dalek actions in the series--making the Daleks the only life force in all universes.
There's an odd bit with Timothy Dalton as the president of the Time Lords. First, most of his acting appears to have been blue screen, which is too bad, because he gave a hell of performance, although it fell slightly flat because of his lack of interaction with almost any other person directly in front of him. He was still great.
There's a neat part of the president's role that's buried: the Doctor calls him Rassilon in the moment when he's broken the link, sending the time lords back to the Time War. Perhaps this the snake eating its own tail, because Rassilon is the founder of the time lords.
Perhaps Rassilon was revived to lead the time lords in the war, just as the Master explains his returned existence to being resurrected by the time lords to fight as a relentless warrior. (Instead, when the Dalek Emperor was in his ascendancy, he fled to the very end of time.)
What's most profound about the last episode are two elements. First, the Doctor is given some grace to have what he calls his reward: he travels around and saves some companions' lives, and aids Donna Noble's financial condition, while finishing up by saying goodbye to Rose before she's met him. In previous regenerations, as I recall the ones I saw, the Doctor has moments, if that. (In one, the previous actor apparently refused to appear in the regeneration scene.)
Second, this Doctor doesn't want to go peacefully. Christopher Eccleston's Doctor wasn't ready--it wasn't clear how long he had even had the current body. But he watched Rose nearly sacrifice herself to save the Earth's future (and probably beyond), and thus was willing to accept his fate.
David Tennant didn't go quietly into that good night. He looks slightly off camera, and says, nearly tearfully, "I don't want to go." And then regeneration happens, violently, seemingly nearly destroying the craft.
I thought for just one moment: did the producers fool us all? Was Tennant going to stay on, become a dark Doctor, provoke a huge crisis? But, no, he had no choice, the Doctor. He wasn't ready to die, and off he went.
And like his predecessors, the new Doctor doesn't remember the existential angst and what-all. He's just a new man with old memories and off he goes.
In the past, there have been interstitial bits of a several minutes long shown after Christmas episodes that explain part of a transition, made to be shown for a UK charity. We may get a few minutes more from Tennant yet, although I doubt it. (Jason Snell tells me that these appear before the Christmas specials, which is right on--he suggests maybe November 2013, the 50th anniversary of the first airing of the show.)
One more thing: many blogs and reviews are saying the Master was destroyed or died when he attacked Rassilon as the Doctor severed the link bringing back Gallifrey and the time lords. I've watched the ending a few times: it's pretty clear that this is an entirely open issue. The Master may have been brought back to Gallifrey; may have died; may be in an entirely different state altogether.
Update, Dec. 19, 2010: I've just read The Writer's Tale: The Final Chapter, an epistolary email account of Russell T. Davies showrunning about two years' worth of Doctor Who, monitoring Torchwood, and being less involved with the Sarah Jane Chronicles. Davies spills the beans about the End of Time in the book!
The mysterious woman is the Doctor's mother. The Time Lord president is Rassilon, with the Time War having destroyed sense so much that he (and many others) have been resurrected by time itself. The Doctor's mother appears to Wilf because time is out of joint, and thus she peeks through the cracks.
Further, he had the last line of Tennant's career ready for some time before the script was written ("I don't want to go"), and always envisioned the Doctor losing this incarnation to save someone perfectly ordinary, but not without railing against his lot first. It's a good read. Starts slow, then picks up. Davies is a fanboy, but tried to balance his desire to nod and wink at the audience with terrific storytelling.