The first series of the new Doctor Who had an arc to it which, in my opinion, has not really been duplicated in the series since. That is not to say that every individual was great -- I would say three single-episode stories and two two-parters are quite noteworthy, with the rest being either somewhat forgettable or seriously flawed -- but the overall sweep of the season, combined with the fact that this was a fresh approach to the series, made it add up to greater than the sum of its parts.
There are three main themes that go through the series:
(1) Getting to know Rose and the Doctor. It's a new Doctor Who, many years after the last, and the audience has to get to know the characters and what the rules of the universe are (though, as always, the rules in Doctor Who are pretty flexible). The Doctor and his new companion have to get to know each other, too, and along the way we also get to know the other recurring characters in the series -- Rose's mother and boyfriend, the Slitheen, Captain Jack, etc.
Related to this:
(2) The Doctor coming to terms with his past, and in particular with the fact that he committed genocide against both the Daleks and his own people. He's an angry, guilty, secretive guy at the beginning of the season, but by the end he's come to terms with who is is and what he's done.
(3) Bad Wolf. At first it's just something that happens, then it keeps turning up, and its final meaning becomes clear only in the final episode.
Most of the episodes, even the ones that aren't very good, have something going on with one or more of these themes. The most interesting one of these to me is the second, but I think the fact that all three are there strengthens the series and makes even mediocre episodes more interesting on re-viewing.
The first big character development episodes are paired nicely in episodes 6 and 8. We get to know about the Doctor's history in any detail, and how deeply traumatized he is, in episode 6, 'Dalek', which I would identify as the first really good single-part episode, in which he ultimately chooses to let go of some of his anger. Then we get to know Rose in more depth a couple of episodes later, in episode 8, 'Father's Day'.
Immediately after that we get to 'The Empty Child'/'The Doctor Dances' (episodes 9 and 10). Ultimately the New Who has gone to the 'little kids are creepy' well more times than I would have liked, but it's used for the first time here to great effect. We also get Captain Jack, the second man with a time machine that Rose has met, and who is the closest thing we get to an actual match for the Doctor, both in power and in coolness, this season. Another thing I like about this two-parter is that there aren't really any bad guys -- Captain Jack has carelessly caused the problem, but he also ends up saving lives (knowing that he might end up dying as a result), the nanobots are just trying to heal everyone as best they can, the kid just wants his mommy, and Nancy conquers her fears to save the day.
But it's also the episode where the Doctor finally gets to finish the healing process that started in the Dalek episode. In the middle of a war (reminiscent, perhaps, of the Time War, in which he ended up destroying his home planet, all of the Time Lords, while saving the universe from the Daleks) he solves the puzzle of what's going on and gets the nanobots to recognize their mistake. If the Doctor had his way every episode would be like this -- there are no enemies for him to defeat, and at the end of the episode he gets to say, "Everybody lives!" It's a great cathartic moment for him and for the series, and while later seasons tried to recapture this I don't think they ever succeeded.
This would be a nice and satisfying place for the season to end, but it didn't, which I think is ultimately to the good. After 'Boom Town', which I don't have a lot to say about, even though I thought it was a pretty good episode (though it is an example of a trope I find annoying, where most of the episode's drama is centered on a moral dilemma, but then the moral dilemma disappears of its own accord and we never get to see the characters actually have to follow through on it), we get to the two-part season finale, 'Bad Wolf'/'The Parting of Ways'.
The ending of this episode is a total deus ex machine -- nothing we have seen previously would indicate that exposure to the time vortex would make you super-powerful, or that the Doctor would be able to save Rose in the way that he does, or that it would end up killing him. (Don't Time Lords have to look into the time vortex during their training anyway?)
This is all redeemed somewhat by the way the episode fits into the series. In this episode we find the Doctor unable to make the sort of sacrifice he used to end the Time War. He can't bring himself to kill all the Daleks if it means destroying Earth too. (Could he have done it if the Earth weren't in play, and it was just him and the other people on the satellite? I think he would have -- nobody knows better than him that the Daleks are bad news, and everyone on the satellite is either dead or is clearly willing to self-sacrifice -- though he would have found it tough.)
And then we get to see the dark side of power, even when it is used for good. If 'The Doctor Dances' shows the Doctor at his most powerful, redeeming himself by resurrecting everyone (and making them better than new in some cases), this contrasts with what happens once Rose becomes Bad Wolf. At that point she commits genocide, essentially willing the Daleks out of existence, which is kind of terrifying and a dark contrast to the 'Everyone lives!' climax of 'The Doctor Dances'. Then she brings Captain Jack back to life, and that's kind of terrifying too (particularly in light of later revelations that she's essentially made him immortal, but even without it, that level of power over life and death is scary). Then the Doctor, who has come to terms with his own life-and-death decisions, saves her, sacrificing himself in the process. His line: "You need a doctor."
These moments would mean a lot less without the development of the themes throughout the season.
To compare with season 2:
The next season still has something of the first theme (getting to knooow you) since we're getting to know David Tennant and some of the other characters become more fleshed out. The equivalent to the 'Bad Wolf' theme is Torchwood, which keeps turning up for no good reason, but this doesn't pay off in the same way, as far as I can tell -- this time it really does seem to be just a coincidence that the Doctor keeps running into Torchwood references. And there isn't really an equivalent to the Doctor's character development in season 2, though looking back on it I think they were trying to do the same sort of thing with Mickey, so he goes from being a relatively ordinary guy who is happy on Earth to multidimensional hero who decides to stay in the other universe. But he's not in all of the episodes and, anyway, the Doctor is the one who we really care about in this show.
(Which is not to say that the second season doesn't have good episodes -- I certainly kept watching -- just that it didn't have as good an overall arc as the first.)