In case you haven't seen it yet, I recommend watching the short prequel: The Night of the Doctor. There's also The Five(ish) Doctors Reboot; probably best to watch that after the main episode.
Anyway, spoilers follow.
This episode certainly started out with a bang: UNIT picked up the TARDIS via a helicopter and flew it to Trafalgar Square. The Doctor had to open the door and lean outside to use the phone (something that's been previously established in the series), so this reminded me of Matt Smith's first episode where he regenerated inside the TARDIS and then flew over various London landmarks while hanging out of the door. This is one of the bits that didn't quite make sense. If UNIT had known that he was inside then they could have asked him to teleport. On the other hand, since they didn't know he was inside, what were they trying to achieve? They'd wind up with an empty TARDIS at the Tower of London (the helicopter's original destination) while the Doctor would be stranded in the middle of the countryside and have to hitchhike. Still, it looked very impressive, and I enjoyed watching it.
The prequel established that the 8th Doctor regenerated into "The War Doctor". So, how does that effect the numbering for later Doctors?
Edit (2015-11-22): Everyone else seems to have stuck with the traditional numbering, so I've changed this post to reflect that.
There have been previous stories where multiple incarnations of the Doctor met up. It's been a while since I saw them, but as I recall they either started with both/all Doctors popping up in the same place or they followed the "current" Doctor until his predecessor(s) arrived. This episode was a bit different, since it gave the 10th/11th Doctors their own separate(ish) stories before they met. I think that helped to establish them as equals, and it also made me think about the time fracture that brought them together. If that hadn't been there, this could have been a simpler story where they work on different ends of the same problem without actually meeting. There's some precedent for that in Star Trek novels, e.g. Federation. In a way, that's a problem, i.e. they didn't both need to be in both times, so one of them was redundant. (There's a scene towards the end where they literally speak and act in unison.) However, I don't mind that: it was fun to see them interact, a bit like siblings bickering.
This also means that we can treat the involvement of the War Doctor as a change in history. Back in 2007 I wrote about "The Shakespeare Code" (the 2nd episode with Martha Jones as the Doctor's companion). In particular, I wrote about multiple "laps" of the timeline: the episode inadvertently implied that Martha Jones had doomed 20th century Earth! (The logic's a bit tricky, so please read that post for my explanation.) Coming back to "Day of the Doctor", the War Doctor didn't need to be there to solve the Zygon problem. His main contribution was to suggest a way for them to get inside the TARDIS-proof Tower of London, but if the Moment hadn't created the time fracture then the 11th Doctor wouldn't have left 21st century London in the first place. Just to be clear, I'm fine with that: the real purpose of getting the Doctors together was to address what happened in the Time War, and I'm satisfied with the War Doctor solving the extra problems that he created. Mind you, I'm surprised that the 10th/11th Doctors were even willing to talk to the War Doctor, based on the ending of "The Name of the Doctor".
More generally, this episode took advantage of time travel to tell the story non-sequentially: we sometimes saw the effect before the cause. (It's raining fezzes!) I thought that worked quite well. It rewards people for paying attention, and it avoids the feeling that the writer is cheating with a retcon. Moffat has done something similar in previous episodes, e.g. the Doctor said something to Amy Pond in "Flesh and Stone" (while she had her eyes closed because of the Weeping Angels) and it later turned out that this was a future version of him from "The Big Bang" (the final episode of that series). I found that I picked up a couple of points on my second viewing that I'd missed the first time around.
On a similar note, the episode reminded me of the computer game Day Of The Tentacle. In that game, you control characters in different centuries, and you need to take advantage of the different ways that time works, e.g. you could zap something from the past to the future instantly (via the toilet bowl in the time machine) or you could bury it and get the future character to dig it up. In the episode, they did something similar with the sonic screwdriver, so that the same program could run as a background job for hundreds of years. I'm not quite sure how well that fits in with continuity, e.g. when the 11th Doctor's screwdriver got partially eaten by a flying shark, but I'll go along with it.
I also liked the brief appearance by the (future) 12th Doctor. It makes sense that he would remember what his past selves got up to, so he had more of a reason to appear than any of the prior incarnations.
Speaking of the screwdriver, the War Doctor explicitly criticised the 10th/11th a couple of times for trying to use their screwdrivers as weapons. However, all of them used their screwdrivers together to fight a Dalek later on, presumably by repelling it magnetically, which does rather undercut what he was saying.
Each Doctor had his own copy of the TARDIS: these were definitely the same machine, but they all looked distinct, so I could tell at a glance which was which. For instance, the 11th Doctor's TARDIS was the only one with the St John Ambulance roundel on the right hand door. That was a nice touch.
Looking at the Zygon story, the Doctor(s) wrapped it up by forcing them to negotiate with the humans, and he made them all forget which group was which so that they'd have to find a fair treaty. However, I'm not quite sure about the logic there. The Zygons established that they copied memory along with appearances (specifically the one who mimicked Osgood and mentioned her sister), so presumably the memory device just knocked out the Zygon memories. That means that everyone would believe themselves to be human, so why wouldn't they all agree on a treaty which was massively biased that way? Looking back at the episodes with the Flesh/Gangers ("The Rebel Flesh" and "The Almost People"), they had a similar premise, i.e. two identical groups who both believed themselves to be the originals.
In case anyone missed it, the whole business with Osgood's asthma inhaler established which of them was the Zygon: only the human had an inhaler (it didn't get copied with the rest of her clothing), but both of them needed to use it. In another context, I could see this being handled more like a werewolf story: the Zygon version would then know that she'd turn into an invader as soon as her memories came back, so she might want to prevent that based on the person she believed herself to be, e.g. by surrendering as a prisoner.
Anyway, I quite liked Osgood as a character. She was quite blatant as an audience surrogate, and I'm not sure whether anyone even makes glasses that big anymore, but I'd be happy to see her in more episodes. As for Queen Elizabeth, it may be worth noting that "I know I have the body of a weak and feeble woman" is a quote that's been attributed to her in real life. (According to Wikipedia, it's from a speech to the troops at Tilbury.)
As for the Time War, I'm really not sure how well this fits in with what we've heard before. The basic idea is that the Doctor did something which (apparently) killed all the Time Lords and all the Daleks, although later on lots of the Daleks came back. However, in this episode he says that if the Daleks knew there were three versions of the Doctor then they'd call for reinforcements. Does that imply that this wasn't the full Dalek fleet in orbit? If so, then whatever actions they took, there would still be more Daleks roaming around elsewhere. Or did the Daleks have allies? Also, looking at the episode "The End of Time", that had a group of Time Lords (including Timothy Dalton playing Rassilon) who put Gallifrey in a time lock at the end of the war, then tried to bring it back later. How does that fit in with the 9th Doctor destroying the planet?
I wonder whether Big Finish will make any new stories with John Hurt, e.g. his "final" confrontation with Davros. In the past they've said that the Time War is off-limits, but maybe that policy will change now that we've seen some of it on TV. Their FAQ says: "The terms of our licence with the BBC allow us to only produce 'Classic' Doctor Who. This means that we can only use the first eight Doctors and their companions. Anything connected to the new series – even characters who are no longer featured – cannot be used by us in a Big Finish Doctor Who production." However, the recent Destiny of the Doctor series includes the 9th/10th/11th Doctors, so I assume that the FAQ is now out of date.
I did like the bit in the prequel where Cassie refused to go with the 8th Doctor because of the Time War. I'm guessing that this was a war with multiple fronts, e.g. the Daleks would invade a particular planet in a particular century and then the Time Lords would turn up and nuke it from orbit to get rid of them. That would explain why Cassie was so hostile; it also allows for time travel in general without it all getting really confusing on Gallifrey itself.
I'm not sure when this story was set in the 10th Doctor's timeline. He recognised the reference to "Bad Wolf girl", so this is presumably after he'd parted company from Rose Tyler, and while he didn't have a regular companion; possibly in between Martha and Donna? The War Doctor didn't have a companion either, so it was only the 11th Doctor who did, and Clara was the one who persuaded them all to find another way rather than blowing up the planet. I've mentioned this before: when he went to Pompeii, Donna nagged him into rescuing Caecilius, whereas when he went to Mars he was happy to stand back and watch everyone die. So, looking at this as a 50th anniversary, it makes sense to emphasise one theme of the series: the Doctor needs a companion.
I did like the promise that the Doctor made when he chose his name:
"Never cruel or cowardly. Never give up, never give in."
That reminded me of the Lions' Prayer (from my LARP faction), and it's a good motto to live by.
As for the ending, I wasn't keen on Tom Baker's appearance. It didn't really make any sense, so I have to wonder whether they even had a script or whether they just let him ramble on for a few minutes. Still, I'm sure that other people were glad to see him.
Overall, I liked the episode. There was visual spectacle, some things that made me laugh, and other scenes that were moving.
I deliberately didn't watch the trailers for this episode until after I'd seen it. Now that I've seen them, I'm glad of that, because they did give a few bits away. Based on that, I haven't watched the trailer for the Christmas episode yet, so please don't tell me anything that happens in that.