It is likely that you will need to contact the particular program director to receive an answer specific to your situation, as a number of elements are specific to the program, the department and the institution or university. However, allow me to respond from the point of view of the director of a graduate school in a Russell Group university in the UK.
Given that you accepted an offer from University A about a year ago, it is likely that you would have turned down the offer from University B or allowed the offer to lapse. Either way, as a year as passed, it is highly unlikely that the offer from university B is still valid. Thus, you will need to reapply.
The administrators of graduate programs, whether operated centrally as mine is or devolved to individual departments, will retain records from previous applicants. The type of information stored and the duration of storage will depend on the individual unit. Regardless, it is highly likely that some record of your information -- perhaps even your full application -- will have been retained, given that only a year has passed.
Thus, I wouldn't hide the fact that you had previously applied, been accepted and declined a place in University B. It's quite the opposite, in fact. Your application should now contain some explanation about the following:
- Why you are reapplying
- What was the reason for your decision to decline the offer
- What have you been up to in the period of time between applications
These explanation should be made on top of the information you'd be providing about your proposal, your qualifications, etc. It might form some part of your personal statement.
Once your application has been received, it will be assessed in the usual manner. A flag will be raised that you'd applied before and (at least in my graduate school) your previous record would be appended to the new application. This forms the evidence on which a decision is made.
You need to be aware that submitting an application in this round that is identical to the one you submitted previously is not a wise move. First, it's been a year or so since you wrote the proposal and the science might have moved forward in important ways. Second, you have had more experience and may have had time to produce research relevant to your proposal. Third, the program or department's staff mix might have changed, so that potential supervisors are now unavailable. Fourth, the program or department's research priorities might have changed so that the areas of research are now different. When crafting your new application, I suggest taking these issues into account.
Contacting program directors is always advisable, at least to receive information specific to your situation. Don't expect them to know your case if you cold call, but if you prepare them sufficiently, they should be able to provide you with valuable advice that you can use to your advantage, even if that advice is "don't bother applying".
Finally, reapplications from applicants who've previously been accepted into one of our programs and who've subsequently declined is unusual, but not unknown. This year, we accepted about 400 or so new PhD students. About 2.5% were from students who'd deferred their entry (that is, they accepted the original offer, but asked permission to delay their start dates by a few months). We had one reapplication, which was eventually rejected.