Please clarify which is UK English, American English, and where and when to use which:
- Fall term (American English?)
- Autumn semester (UK English?)
- Autumn term (wrong?)
- Fall semester (wrong?)
Fall and autumn are famous differences between British and American use, both recognised in all forms of English, but fall almost never actually used in Britain and Ireland and rarely in Australia or New Zealand, autumn almost never used in America, and Canada using both.
Term and semester are not strict synonyms. Term applies to any part the academic year is broken into, while semester strictly applies only to a bipartite system, with trimester (yes, the same as pregnant ladies have) for a tripartite and quadmester or quarter for a quadripartite system (such as Australian schools have).
To add confusion, if you have three normal terms, but also offer supplementary classes over the summer, then the terms can be referred to both as trimesters (taking the name from the three only) and as quarters (taking the name from all four).
Not long ago you could be pretty confident that a given US college or university would use semesters and a given UK or Irish one would use trimesters. In the last few decades though, many UK & Irish colleges introduced a policy of semesterisation (such a cumbersome word to put on posters at Student Union protests, but put it on, we did) and now have semesters while some US colleges have three-term or three-terms-with-supplementary-summer-term systems.
Since they aren't perfect synonyms, there are times when precision leads to US speakers using term or UK speakers using semester or trimester as appropriate. At times when one could choose freely between them, a US speaker is more likely to use semester while a UK speaker to use term. However students or faculty at UK colleges with semesters are slightly more likely to use that term, given its novelty to the college and that the move toward semesters was a controversial one in many colleges, resulting in the word being strong in people's minds. It is also common for such UK colleges to call them semesters in timetables and official literature, but people refer to them as terms informally.
The names of individual terms in the UK & Ireland differ from college to college. For those with semesters either Autumn and Spring or Autumn and Spring and Summer are common, or they might just be Semester 1 & Semester 2. For those with trimesters, some examples include:
Michaelmas, Lent & Easter; Michaelmas, Epiphany & Easter; Michaelmas, Lent & Summer; Autumn, Spring & Summer; Michaelmas, Hilary & Trinity; Martinmas, Candlemas & Whitsunday.
Traditionally, the English split the year into Hilary, Easter, Trinity & Michaelmas with Ireland following suit (in some ways the history of Trinity College Dublin is as much a British one as an Irish), while the Scots split it into Candlemas, Whitsunday, Lammas & Martinmas. As they each altered their systems in different ways at different times, they ended up having different names from each other.
These sometimes survive into semester names, with e.g. St. Andrews having Martinmas & Candlemas semesters, due to their once having a trimester system with the terms named, Martinmas, Candlemas & Whitsunday.
Edit, mplungjan's comment makes me consider that some of these names may be unfamiliar to some.
Easter: Probably known to all, but included for completeness. The movable feast of the resurrection of Christ. First sunday after first full moon, after 21 March, it can be anywhere between 22 March and 25 April.
Lent: The forty days before Easter.
Michaelmas: Feast of St. Michael the Archangel, 29 September.
Epiphany: Feast of the revelation of God made flesh through Christ, and the recongition of this by the magi. January 6.
Hilary: Feast of St. Hilary of Poitiers, 13 January.
Whitsunday: Generally means the celebration of Pentecost (descent of the Holy Spirit upon the apostles) celebrated on the seventh Sunday after Easter. In Scotland, there was a period when it also refered to a fixed holiday, 15 May.
Trinity: Trinity Sunday is the first Sunday after Pentecost.
Lammas: Feast of St. Peter in chains. August 1, combined in Britain with the Anglo-Saxon hlaf-mas (loaf mass) and absorbing the Lughnasadh of Scotland, Ireland and Man.
Martinmas: Feast of St. Martin of Tours, 11 November.
American usage for September to December in this context is "Fall", never "Autumn". [Correction: a commenter who did more work with Google has found counterexamples. Fall is more common, but not universal] Term is apparently used at Harvard and Princeton, but I believe Semester (for a term of approximately 15 weeks) and Quarter (for a term of approximately 10 weeks) are more common. Not all our universities are so long in the tooth!
I defer to UK readers for usage there.
The British equivalent of American semester is term. Schools have an Autumn Term, a Spring Term and a Summer Term. The University of Oxford has a Michaelmas Term (autumn), a Hilary Term (spring) and a Trinity Term (summer).
Term is the hypernym in this context. University calendars are typically based on a quarter (10-week) or semester (15-week) system. At either institution, though, the expression winter term could be used when referencing the winter quarter or the winter semester.
Some universities might offer courses in a condensed format over the summer, in which case, you might hear a professor say something like, "I'm only teaching in the second short term this summer."
Term can also be used in the context of elementary and secondary education: "My daughter got a B in science last term."