At the University of Cambridge, examinations took the form of oral disputations, in which students were permitted to participate after a minimum of two years of study (less if one was the son of a nobleman). Four such oral exams were required to graduate, with the student serving twice as opponent and twice as respondent. Needless to say, everything was conducted in Latin.
All of this was tied in to a wider series of regulations, which we know about from a statute of 1390 outlining the lecture course for the masters degree. A bachelor student was required to hear lectures for at least three years – he would be expected to specialise, but there was a core curriculum in which Aristotle featured heavily (Posterior Analytics, Physics, Ethics, and Metaphysics were all on the curriculum, as were Euclid's Elements, Algorismus, Compotus and Tractatus de Sphera). Students were further required to have heard lectures on the Trivium (grammar, rhetoric and logic), the Quadrivium (arithmetic, geometry, music and astronomy) and the three philosophies (natural, moral and metaphysical). Students were expected to attend "ordinary" lectures, which sat at the heart of the curriculum, but there were also "extraordinary" lectures available, often delivered by guest lecturers.
After the requirements of the first two years of study had been satisfied, the scholar reached the stage of sophista generalis, which was the midway point between joining the university (matriculation) and graduation with a bachelor's degree. At this point he would begin preparation for the examination process.
The first, preliminary step in the exam process was known as "responding to the question." We have only one late source detailing what this involved - an account by John Stokys, which is outside the period you specify (mid-sixteenth century), but there is no reason to suppose it does not describe the way things were done in the earlier period. According to Stokys, the response took place before Candlemas (2 February), starting at 9am, and the scholar being examined would gather fellow students from his college and the master of the college would lead them to the Schools building for the exam.The master, denominated "father", presided as respondent. He sat, and the scholar, denominated "child", was given an academic hood and stood to answer questions. This stage of the process was not really a serious exam – it was often prefaced with some humorous verse, and scholars almost invariably passed; their colleges would not have allowed them to proceed to this stage if they were not confident they would do so.
Passing the process of "standing as a questionist" entitled a scholar to proceed to an act of determination, and it was this that resulted in the award of a bachelor's degree. This process began in the week before Quinquagesmia Sunday (the Sunday before Lent), and all questionists would gather in the Divinity School between one and five in the afternoon to be orally examined by their masters. The individual exams can't have lasted very long, as there were a number of students being examined, and there was a one hour break in mid-afternoon for a meal. On the last day of the week, presumably after some discussion among the masters, the suitability of the scholars was agreed on and pronounced orally by two regents appointed by the university. Those who passed this stage entered a "period of determination", beginning with a church service and followed by the four disputations, held in the Philosophy School.
Disputations took the form of formal arguments arranged at times dictated by the university. Unfortunately no manuscripts survive at Cambridge to show the exact content of one of these disputes. The general purpose, however, was to allow a scholar to demonstrate the extent of his knowledge and his command of key arguments.
Disputations were public events and they took place before an audience of other students. They were much more solemn affairs than the "responding to the question" stage. For the first two debates, the scholar sat (in the later part of the period he did so on a three legged school, which is why the Cambridge exam system is still known as the "tripos") and debated with the master of his college. He propounded and disputed two questions on this occasion, and the debate was followed by what amounted to a "masterclass" - a debate between his master and two regent masters, who responded. There was then a five week break, and the exams recommenced after Lent.
The form of this second series was different to the first. Each scholar ("determiner") taking the exams was accompanied by a junior sophista and had set before him three written questions on dialectic and philosophy. He then acted as respondent and debated with any scholar or bachelor who cared to turn up and argue against him. A third man, usually a graduate, served as replicator and summed up. The process was supervised by proctors.
This final examination took place on the Thursday before Passion Sunday (concluding 12 terms of study). At the end of it, presuming things had gone well (which, once again, they almost invariably seem to have - there are records of only 26 failures between 1454 and 1500), the senior proctor would declare: "By the authority which we enjoy, we decree, appoint and pronounce all the determiners of this year to have finished their determinations, and now be bachelors of arts."
Much the same syllabus was followed at Oxford, where we have "graces" describing the balance of the course. This could vary and students were expected to specialise. A sample grace lists one term of astronomy, three of moral philosophy, and two of metaphysics.
Source for this: volume one of A History of the University of Cambridge, by Damian Riehl Leader.
To become a Bachelor a student merely had to have studied under a master for the required number of years, and to have listened to the required sets of lectures on the right texts. There were no formal examinations. A statute of Oxford University from the early 14th century declared that no one could be declared as a bachelor, "unless he has confirmed with his own oath that he has heard lectures in arts for four years in schools where there is a studium generale". We can see here that attendance at lectures was not even requisite to be confirmed by the masters, students were merely required to swear under oath that they had heard the lectures. The lectures were based around Aristotle and other ancient authors, with a set list of texts which did not change (including Boethius, Priscian, Donatus, Porphyry, Euclid, and John of Holywood). The lectures were split into ordinary and extraordinary lectures, with the former being lectures on the essential texts, and the latter the additional texts. The curriculuum consisted of the seven liberal arts and three philosophies.
However, to be licenced to dispute, teach, or practise for oneself, and to become a master or doctor was much more onerous and few students applied for this.
To progress to the higher level of licentiate, master, or doctor, a Bachelor had first to study for a further period, in Oxford this was three years. During this time the bachelor would have to give cursory lectures of his own to students, probably no more than reading out the text and giving a short commentary. But during this period they were also examined by carrying out a series of disputations with peers or a master. These followed the questiones format, which would have been familiar to them from their lectures. In the lectures the master would read the required text to the class and then extract a word or phrase, for which a number of different interpretations would be proposed and then refuted, before the master settled on his preferred answer. The Bachelor's disputation would follow a similar format, in which, under the master's presiding role, they would oppose, or provide their given answer, which would then be responded to by others invited to do so by the master. There would be several rounds of back and forth before the master declared an end.
For Oxford, it was expected that a bachelor take part in four disputations with masters during this three year period of study, and two per year with his fellow bachelors. At the end of the period he would be asked to perform an ordinary lecture before a group of masters who would judge the bachelor on their performance and ability to handle the ensuing dispute.
A successful lecture and disputation would earn the bachelor a license, and the licentiate would then be allowed to incept as a master. During this short period between license and inception the licentiate would give two more disputations (one acting as a proponent, another as a respondent). They would then have a feast and the licentiate would become a 'regent master'. For the rest of that year and for the next year, they would be known as a 'necessary regent'. This meant they were required to give lectures and attend disputations whenever required. Only after this time had passed were they allowed to move on to study in a place of higher learning, if they wished.
Source: The University of Oxford: A History, by L.W.B. Brockliss (2016), published by Oxford University Press.
Finally! A question in my field! And then I see others have already answered this quite capably. I would add to the two previous answers thusly: They are reporting on Oxford and Cambridge, but I believe the model they are describing would fit most western European medieval universities.
The administration of examinations & disputations by faculty conforms to the "guild" model of organization/governance. They exercised the power to determine entry into their own profession, as passing the exam earned the candidate a licencia docendi, a license to teach.
Source: Domonkos, L.S. "History of Higher Education"