I am applying from Europe to the U.S. for tenure-track positions. One of the advice that I heard from my colleagues is that once one offer is received, one should contact the other universities that one has a preference for, and tell them about the offer. However, I do not understand how this act of telling another university about an offer could actually put any pressure. What does one say in such a communication? And what is one expecting from it? (For example, if all the interviews are already scheduled, one can hardly expect the university to "hurry along" and pick a candidate...) I am interested in hearing the answer for American institutions, but as long as the region is specified, I should like to hear the European versions as well.
Simply say that you are writing to let them know that you have received an offer from institution X. If there is a deadline for you to answer the offer, tell them what it is. This may be helpful for the department chair to use in pressuring their administration or other involved parties to make a decision sooner than they would have otherwise - yes, people can often "hurry along" when the circumstances warrant it. It may also have a psychological effect of making you look more attractive and desirable, especially if institution X is a prestigious one. But let them be impressed by the news itself; keep the tone of the email neutral and informative and avoid coming across as smug or entitled-sounding.
As for what to expect, that seems beside the point. Just like you would update the places you've applied to if you had a paper accepted to Nature since that would make you a more attractive candidate, you want to let them know about a job offer from another institution. It may or may not have any positive effect, but it sure can't hurt.
If it matters (I don't think it does, but you asked), I'm in the US. Good luck!