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As universities provide short deadlines for faculty position offers, applicants can accept an offer-1 (verbally or by signing a contract) and which is not the best. What is the recommended option if a late and a better offer-2 arrives after accepting offer-1?
This situation should not actually arise if you are handling your faculty job search properly. If you accept an offer, you should withdraw all your remaining job applications. Otherwise either you are wasting their time in considering you for a position you won't accept, or you were insincere in accepting the previous offer. If you aren't comfortable withdrawing your other applications, then you aren't comfortable accepting the job. You can negotiate on this point, for example by telling them that another job would solve your two-body problem and you hope they can wait on a final decision until you hear about that job, but there's no guarantee that they will agree.
The basic ethical principle here is honesty: you shouldn't give someone a decision they understand to be final without actually meaning it. By default, job acceptances are considered binding decisions in the parts of academia I'm familiar with (certainly in mathematics in the U.S.), so you can't just assume that of course they knew you might change your mind. If you have any reservations or conditions, you should make them explicit before accepting the position. This can't hurt you if nobody really considered the decision to be final in the first place, and it will avoid unethical behavior if they did.
Even though this shouldn't happen, people do occasionally get themselves into this situation. If you unilaterally rescind your initial acceptance and take the other position instead, you face almost no legal risk, since nobody's going to try suing over this. However, you can hurt your reputation, which is a serious danger.
Instead, the way you should handle it is by careful discussions. Typically, University 2 will let you defer their offer for at least a year, since otherwise they look like jerks for trying to steal you away from University 1 after you already accepted an offer. (Another possibility is that University 2 had no idea you had already accepted and will rescind their offer upon learning this.) Then you approach University 1 and apologize profusely for inadvertently creating a terribly awkward situation. You explain that you are willing to come to University 1 and fulfill your obligations, but you have an offer from University 2 and you would very likely leave after a year to go there, so you wonder whether there is any chance they would release you from your acceptance. If they agree, then you are ethically free to accept University 2's offer immediately. (University 1 still won't be happy with you, so you shouldn't do this unless it really matters to you, but asking them for permission is much better than just announcing you aren't coming.) If University 1 insists that they need you next year, then you defer University 2's offer and show up at University 1.
But you really shouldn't let yourself get further faculty offers after you've already accepted a job. You might be able to get away with it once by explaining that you accidentally forgot to withdraw your other applications, but you really don't want to develop a reputation over time as someone who deliberately manipulates the system in unethical ways.
There's a missing aspect in both of the other answers here: how long did the first university give you to make your decision?
First, you should immediately acknowledge the receipt of their offer, and then if U1 gave you a week (or a day, it happens!), the you should tell them immediately that you like their offer, but that you have other applications pending and that you need more time to decide. This starts a negotiation and recognizes to them your continued interest in the position. Then you should get in touch with all of the other universities which you would still contemplate accepting an offer from and let them know (without, necessarily, naming U1) that you have received an offer with a short deadline to accept. This gives the other universities where you have applied an opportunity to communicate their ongoing interest in you (or not) and help you make your decision about whether to accept or reject U1's offer before hearing from the others.
You should keep this communication up as things progress. There are so many things going on behind the scenes that you don't know about. The more open your communications are, the more likely you are to find the best position for yourself and allow the places you don't go to find their next best option.
When a university makes you an offer, that means they have decided that you are the best candidate that applied that they think they can actually get. They have made a strategic decision to offer you a job over other applicants. They have given you some sort of time limit to decide, because they have other options still waiting, and they don't want to lose their chance at them if you are going to decline. This is true of the places which you haven't heard from yet as well. Letting them know you have an offer will push them to figure out what they want to do.
As others have said, once you have accepted an offer, you really should bow out of the other positions you have applied for, but you don't have to let it get to that point if you communicate with everyone well.
Although some would argue about the ethics of the situation, I'd say accept the better offer... unless acceptance of offer-1 was in the form of signing a contract. My reasoning is this: If you were already employed at Uni-1 and Uni-2 made you a better offer, you'd work out the duration of your contract and move.
If you have signed a contract, you are ethically and legally bound to honor it. (Whether anything bad happens if you break it is another question.)
What you must not do is accept Uni-2's offer until you are clear of Uni-1. There's a certain amount of peril in that because Uni-2 might do the same thing you are contemplating, namely withdraw their offer for a better prospect.
For my understanding it is totally okay to turn down an offer even after verbal acceptance, as long as you didn't sign the contract. So I assume we are talking about an offer which you accepted in writing (signed). Ethically speaking, the point when you sign the contract should be the point when you inform other universities that you withdraw from their searches. If you already have signed the contract with University 1, but you really want to accept offer from University 2 for X,Y,Z reasons, then there is nothing wrong in doing so. But you should be aware that you are most likely will burn bridges with University 1, since you crossed an ethical line which most of the Universities at least want to see being honored by their future employees. University 1 also told other candidates that the position is taken, but in reality, that doesn't make a big difference, if not a lot of time passed between signing the contract and you declining the offer. I would be as honest as possible, apologize, tell them about your reasons. If you have good personal reasons, most of the people would be understanding. Professional reasons might raise some temporary or permanent anger on their side. However, eventually they will let you do this, because they don't want a faculty working in their department who only is there because he legally has to be there. You can do decline in writing or via phone. I would use the way they were communicating with you most of the time. And remember, by the end of the day, it's your gut feeling which decides. Don't let the other people scare you. Some folks are better listening to their gut, others aren't that good at it. It is good that you are competitive enough to get more than one job offer, and also Universities trying to get good people on the hook. And for the future you might consider a better negotiation ethics. For example it would be totally okay to inform University 2 that you have a written offer in hand from University 1 and you would like to know what your chances are. As long as you don't sign, it's not a massively big deal for any of the participating parties. You might get some good vibrations from University 2 and then it would be up to you whether or not you want to gamble or be on the safe site. But that's life, sometimes you just have to make decisions which don't always cover all you necessities.
I must disagree with "anonymous Mathematician". That advice might leave you without any offers. His/her advice does not account for the fact that campus interviews, and therefore offers, often occur weeks and even months apart. If the first offer is close enough (2-3 weeks) from an anticipated second offer, you should try to ask for an extension. Inevitably, a school will eventually need an an answer and it is perfectly acceptable for you to accept the offer from School #1 while waiting for a campus interview from School #2. You may have to call School #1 back and let them know the bad news, but don't let people on here scare you into thinking you HAVE to stay with School #1. In my experience, School #1 is perfectly understanding. They would rather let you go than have a bitter faculty member who is disheartened that they didn't have the foresight to reject School #1's offer.
