Typically they have elaborate constitutions, and are equipped with:
(a) A Chancellor, who is a ceremonial head with no function in the administration of the university, but who is expected to add lustre to it by his fame, and who may be expected to assist with fundraising efforts, or who may even be a substantial donor. Sometimes it’s a minor royal; Princess Anne is Chancellor of several universities. Sometimes it’s a celebrity; the actor Jeremy Irons is chancellor of Bath Spa University (which has a particular strength in the creative arts).
(b) A ceremonial assembly comprising all the graduates of the university, or all the Masters, with a name like “convocation” or “congregation”. It too has little function in running the university, although it may elect or confirm the Chancellor. It usually elects some members to the Council of the University (see below).
(c) A Vice-Chancellor, who is the chief executive and the senior academic in the university. In Scotland he’s often called the Principal.
(d) In some cases, a Visitor who is an independent outsider who arbitrates disputes that can’t be resolved within the University. The visitor’s position is usually held ex officio; i.e. you don’t get to be visitor because they admire your wisdom, but because you are the Queen, the Lord Chancellor, the Bishop of Durham.
(e) An executive body called a Board, Council, Court or similar which consists of academics, often student representatives, sometimes civic representatives, maybe nominees of a bishop or a religious order if the university is a religious university. It will be dominated by academics, and there’ll be an election/nominations system to ensure that the academic representatives are drawn from across the range of factulties/schools/departments into which the university is divided. This is a very significant body, meeting regularly. The Vice-Chancellor reports to it. They hire and fire the Vice-Chancellor and other executives, and they control finances. In the older Scottish universities, this body (called a “Court”) also includes, and is chaired by, a Rector, who is directly elected by the students of the University. The Rector is usually from outside the university and is often a politician or a celebrity; Stephen Fry, for instance, has been Rector of the University of Dundee, while the current Aberdeen Rector is a senior figure in the Scottish Green Party.
(f) A larger body with a title like “Senate” which is dominated by academics and which has final responsibility for academic matters. Professors of the University may be appointed or confirmed by the Senate, which also approves the awarding of degrees. It approves the establishment of degrees and degree course, oversees the conduct of examinations, etc. It usually nominates some of the members of the Council.
Nomenclature is very variable, so e.g. the “Council” in one university may be a small executive body, but in another could be one of the larger assemblies.