This is anecdotal, about Oxford in the period 1970-2000.
Being a mathematician I did not use this code: we expressed our examination marks in the form N alpha^m beta^n, meaning an overall points total of N, with m questions almost complete, and n questions with substantial answers. However, I lived among those who did use the code, and think I understood it in broad outline.
Basically the code was developed in the first half of the 20th century. At that time students in Oxford were classified in their final examinations as gaining Honours in the First, Second, or Third Classes. (There were also special cases: a Fourth Class; one might not get Honours but be awarded a Pass and so get one's degree; or one might Fail.) Basically alpha means First Class, beta Second Class, gamma Third class). But although the final consolidated mark was I, II, or III academics cannot resist making more and more refined distictions and so a more elaborate system developed.
A further elaboration was that as a further refinement the final + or - might be preceded by a ? sign: so for example beta +?+ is better than beta + but not as good as beta ++.
I think that sometimes alpha + etc were used, but purists insisted that it was impossible to be more excellent than alpha.
I do not think that delta was used seriously: as I recall it "s." [ i.e. satis] denoted "Pass" and "n.s." denoted failure.
I think that a mark alpha gamma really signified that the examiner could not decide whether this was brilliance or nonsense: such a mark would have to be resolved to something on the scale before being used.
You ask for the criteria for achieving these marks. That was not how academics at that time operated. Each student was somewhere in one of the Classes: the examiner just had to discern where, and allot the correct symbol. But do not think that this was a random process. When in the late 20C serious studies were made to discover how consistent the assessment was in given subjects, it became clear that there was a very high degree of agreement between different markers.
In Finals a student would do usually eight 3-hour written papers, answering in each perhaps four questions from a menu of perhaps a couple of dozen. Each paper would be read and independently given a mark by two examiners; if there was a discrepancy the two examiners would discuss and produce a reconciled mark.
The whole panel of Examiners (six or eight) of a subject would then meet and produce the Class List. By the eighties there were usually overt conventions about how they would use the marks of the eight papers to do this. But there were a variety of conventions. However they usually looked at the overall (dare I say average?) picture and then required for a First a minimum number of leading alphas and a maximum number of low scoring papers; and similarly for the other classes.
A disconnect with reality occurred when the University divided the Second Class into II-i and II-ii, and the decline in the number of III awarded, and subsequently the decline in the number of II-ii. Although the system may still be used informally all official processing of examination marks in now (I believe) done in terms of University Standardised Marks. This is a 100 point scale where, to comply with national protocols, 70 signifies a I, 60 a II-i, 50 a II-ii, 40 a III, 30 a Pass.