In United Kingdom politics, the doctrine of individual ministerial responsibility is part of the constitution1 by convention.
This doctrine states that a Secretary of State is responsible for the success of his department. Ministers (both Secretaries of State and junior ministers) are responsible for the conduct and success of themselves, their subordinates and their civil servants. Therefore, should the department fail to do its job, the Secretary of State responsible for the department is obliged to resign2. Similarly, should a minister or his/her underlings break the law, or act contrary to the written or unwritten rules of Parliament, the minister must resign.
This is the doctrine which covers the abject failure, sleaze and scandal which make politics so interesting. It seems fairly rare that a Secretary of State resigns purely based on the fact that they cannot do their job. Cabinet reshuffles are generally used instead. Rather, ministers resign under this one because they have broken the written or unwritten rules of Parliament (generally, money in the former case and sex in the latter). Essentially , ministers resign when corruption or undue influence comes to light; cash for questions, for example, or, in the case of Peter Mandelson's first resignation, an undeclared loan3. Ron Davies, the married former head of the Welsh Assembly, broke the unwritten rules of Parliament when he was (allegedly) robbed whilst (allegedly) cruising on Clapham Common, and resigned shortly afterwards. Stephen Byers (former Secretary of State for Transport) resigned May 28, 2002, also under the doctrine of individual ministerial responsibility. In his case it was due to the behaviour of his subordinates, Jo Moore and Martin Sixsmith (talking about burying bad news on September 11, 2001 and the day of Princess Margaret's funeral) and his alleged lying about it, as well as his inability to sort out the railways and Railtrack.
Of course, the problem with all this is that it can create a series of scapegoats in difficult jobs, with ministerial resignations being used as an alternative to actually solving the problem (like constantly changing the captain of a failing sports team). If ministers always had to resign when their departments 'failed', departments such as health, transport, education and the Northern Ireland Office would barely keep ministers for six weeks at a time. Another problem is that it makes it easier for the media to demand resignations for the slightest misdemeanor, as quite a lot comes under the unwritten rules of 'conduct befitting one's position'.
It can be compared to cabinet collective responsibility.
1 Britain does have a constitution. It's just not a written one.
2 Resignation in this instance means resignation from the government (also known as the cabinet or front bench), not resignation from Parliament. A Member of Parliament, once chosen by constituent, remains their MP until death, general election, conviction of a crime or becoming an employee of the crown. This is regardless of promotion to or sacking from the front bench, or joining or leaving a party.
3 Parliamentary rules on matters of corruption are clear, but not exactly logical. MPs and ministers are allowed to accept loans from friends, gifts, free stays in hotels, places on the board of companies and even cash in large brown envelopes, as long as it is all declared on the register of MPs' interests. As long as these are declared, it is held that influence and conflicts of interest should be easy to spot, and so everything is above board. It is if these are not declared, but found out later (as with Mandelson's loan) that it is considered corruption. These things are all investigated by the Select Committee on Standards and Privileges. MPs can also fall foul of the more general instruction to act in accordance with one's position, which is when cruising on Clapham Common becomes a bad idea.