Discussion on this has already started elsewhere, with some talk on the Trump thread and a thread having been created devoted to one specific candidate, but this is definitely something that will need its own thread eventually, so why not start it now?
As of right now, the obvious questions are:
Who do you think will be running?
What are the pros and cons of likely candidates?
Who do you want to win?
Who do you think will win?
What are their chances against Trump?
Personally, I think the three people who are both very likely to run and have some potential to win the candidacy are:
Joe Biden, Kamala Harris, and Cory Booker.
Out of those, I would prefer Biden, but as Doods has pointed out on the Trump thread, age is a real factor there. If he won both the primary and the general election, he would be 78 by the time he came to office, and at that age it doesn't really matter how healthy he seems because he could deteriorate very quickly. There would be a very definite sense that he'd only be able to serve one term at the most, and that could hurt his chances. Age aside, though, I think he'd do very well - he has a genuine charm that people respond to, but does also manage to exude a good sense of both intelligence and capability, and there's no doubting he's experienced. I think post-Trump, people will be looking for stability, and he's a rare candidate that could both promote a sense of stability but also get people excited, a combination it would be very hard to compete against. But age really will be a factor - there's a solid chance he just won't be able to run, and it will definitely be something people will be concerned about. Of course, if he did win the primary, I don't think Trump (who, despite being slightly younger, definitely wouldn't be able to use the age card) would stand a chance.
I don't actually know much about Harris or Booker. Both seem perfectly acceptable, although I'm sure both have their issues. Sellers has explained some potential issues with Harris on his thread regarding her, although personally I think even taking those into account she's an acceptable candidate. I think she'd have a hard time against a regular Republican (but even then definitely could win), but I think she'd beat Trump pretty handily.
I know next to nothing about Booker, besides that a lot of people seem to like him but another lot of people seem to dislike him, and that he was a fireman. Personally there is something about him that rubs me up the wrong way, but it's hard to put a finger on. I do think he might exude a bit too much of a sense of just assuming he'll be president one day, which a lot of people will find off-putting, but that might be more the way elements of the media talk about him than anything he's doing himself. Much like Harris in the general election, I don't think he'd be a definite winner under normal circumstances but in this case I think he'd be a safe bet.
The Obama camp have apparently been pushing former Governor of Massachusetts Deval Patrick to run. I don't know much about him, but some pretty basic research digs up that he's actually got a couple of controversies attached to him that I think would instantly sink a presidential campaign if a rival candidate started using them, and thus I don't see him as a possibility.
There will also probably be a Sandersite candidate who could potentially do pretty well, although it's not obvious who that is yet. The two obvious names are Elizabeth Warren (although she's said she won't) and Bernie himself, although actually I wouldn't expect either of them to do that well this time around, as their main rival will either be a much fresher, quite likely ethnic-minority face, or Joe Biden, both of which I think will bring attention to the issues with Sanders' politics. Harris or Booker would both likely manage to give off a better sense of modernity, which I think young people will actually respond to more if it's offered, and both - being relatively young, relatively black and (although one would assume this will matter less if it's Warren running) in Harris' case a woman - will bring out the worst of the Bernie Bros, whose actions - whether one thinks it's fair for Sanders to take the blame or not - will almost certainly have a negative impact on public perceptions of a Sanders or Sanders-successor campaign. And if Biden's in the race, I think his down-to-earth nature would definitely show up the professor-ish air of either Sanders or Warren.
If Sanders or Warren did somehow win the primary, I think the election would be close. I don't believe Sanders was more likely to beat Trump than Clinton - quite the opposite - and I certainly don't believe he'd do better than Biden, Harris or Booker. Warren I think would get a similar result - the only real difference with Sanders is I think she'd manage to court a higher percentage of younger people but I think she'd lose a lot of what support he had among working class people.
What might stand a better chance in both primary and general election is a younger successor to Sanders (by which I mean someone in their late 40s or 50s, not a Wild in the Streets situation), although I can't think of anyone who could be this. Even then, I think a safer candidate would win the primary.
And finally, a total wildcard who, if he's serious (although I don't think he is), I think could actually do surprisingly well: the Rock. It's hard to tell if he was joking, and even if he wasn't I think if all he had to say for himself was the usual "I'm not a politician" stuff celebrity candidates usually resort to then he probably wouldn't get very far, but he's got charisma and social media skills and a lot of people like him, so if he genuinely puts a good case for himself forward, with the way things are right now, Dwayne Johnson could stand a chance.