A Public Inquiry does just what its name suggests: it investigates an event or series of events in hearings that are open to the public. At the conclusion of its proceedings, it produces a report that anyone can read.
Public Inquiries first came into being as the result of a law that was passed in 1921: The Tribunals of Inquiry (Evidence) Act. They were designed to be more powerful and more independent than anything that had previously been available. Inquiries ordered under the 1921 Act have to be approved by both the House of Commons and the House of Lords, but they can consider any matters they consider relevant to their overall job. A further law, passed in 1977 (The NHS Act), provided the Secretary of State for Health with the right to order a Public Inquiry into the activities of any NHS body or employee. Such Inquiries can be ordered without seeking Parliament's approval, but they have to be limited NHS issues.
When an authority (either Parliament or the Health Secretary) orders a Public Inquiry, it must do two things: appoint a Chair, and provide a precise definition of what the Inquiry is to consider (its "Terms of Reference"). Once someone has been put in charge, and told what to investigate, the Inquiry should be completely independent of those who have ordered it. The Inquiry is expected to devise its own procedures and appoint its own staff.
Although the details will always be complicated, the Terms of Reference with which most Inquiries are supplied boil down to two things: finding out what happened and, if it finds that what happened should not have happened, making recommendations to stop it happening again.
The Chair of an Inquiry is very often from a legal background (a Judge, barrister, or legal academic). He or she heads a Panel that is in charge of hearings and, ultimately, findings. The Panel will commonly be made up of experts in the field in question. This means that, in healthcare Inquiries, the Chair often sits on a Panel with doctors and nurses.
The Panel will be supported by a large team of staff. Solicitors, administrators, and experts will be employed to assist the Inquiry in pursuing its investigations. Public Inquiries generally have such broad Terms of Reference that it would be impossible for the Chair and Panel to sift through all the available evidence and identify all the potential witnesses. This is why they rely on a team of experts to conduct investigations, and bring all the important evidence to their attention.