There seems to be a common idea that banks need your written permission to look into your past. That's simply untrue. I'm not a lawyer, but the idea simply does not survive any real scrutiny does it ?
The law varies from place to place, but banks have a legal duty to check that you are a "fit and proper person" to handle serious amounts of money. Of course a lack of bad evidence does not mean you are honest and competent, but they have to at least show that they tried to find out.
Often this is outsourced to firms like Kroll who get the business mostly becauase the banks has heard of them, and mostly because they do it cheap. I cannot promise that the background checking outfit will obey all laws, but I will say that their fees are simply too small to make organised law breaking worth the effort.
Who are you Banks will often want to make sure that you are the named individual, and that you have the legal right to work for them in that country. There is no point going through the checks below and proving that "X worked at Y firm and studied at Z university", if they don't know you are X.
University The whole point of the university examination system is to inform others of how well you have grasped a subject. Of course there are many universities and they often get really quite odd legal advice, so it is quite possible they will go through a phase of refusing to give it out on an effectively random basis. They also vary in how much detail they will give out. Headhunters are also supposed to check your background. I will say openly that it is easily the most pointless part of the recruitment process for me. The referee is chosen by the candidate, so is the person they think will say nice things, and will usually say you were a good person, even if they have just fired you because of your inability to work with them.
Credit Checks Most large firms have a credit check done on you. It is cheap, and may catch some evidence of "suspicious" activity. Many newbies have incurred debts and students are known to have a rather relaxed attitude to paying bills, so minor glitches are rarely a problem. There is a reasonably good system for international credit checks as well. They are less relaxed about when the courts are involved, and you may need to settle debts. If you have an issue here, then as soon as you get an offer, you should tell your HH and then HR. It is much better to tell than be found out, so that you can put your own spin on it.
Media Check Some background checkers search your name in various newspapers and other media. This is a highly noisy process since names are far from unique. Google on "Dominic Connor" and most of the first couple of page are me, but Star Trek is in there too...
Previous Employers Most contracts of employment explicitly tell you not to ever give any form of reference ever. Banks have been sued for serious money for giving bad references and your employer does not want to be sued because you said nasty things about the idiot who used to work with you. HR departments typically give minimal references with dates of employment, possibly a job title. No permission from you is need for this check.
Gaps Banks ask for explanations of gaps between jobs. They are an indication that leaving an employer was not your choice, and that needs to be covered. Typically they want an explanation of anything above 2 months or so.
The most important thing to know ...is that HR is trained to get very interested if they think you have something you are trying to hide. Even before I became a HH, I had managed people and knew that a larger % of people have "interesting" sections of their career than you might think. Do not try to lie. Although the process is far from perfect, lying about when one job ended and the next begun is likely to cause serious pain, and most probably loss of the job after you've started, which creates another hole, of an even worse kind. Being fired for lying on your resume does not look good to future employers. Many things about your employment are on records such as tax, company directorships, shareholdings and of course court cases. Much of this is publicly acessible to some degree.
In these troubled times, being fired is not such a badge of shame, and a large % of people have had various forms of involuntary career transition. Do not be tempted to hide things that go wrong. You don't need to use big fonts, or draw attention to it, but be prepared to honestly and carefully answer questions on this so that you present your case well.
If you have a gap, even if it is not "suspicious", you do need to show that the time was not wasted. Study, read papers or textbooks, write some piece of software or something. The important thing to show is that you did not sit around watching daytime TV waiting for the phone to ring.