If you look this word up on Wikipedia, you will find a straightforward description of the term as used to describe a display of art or other objects at a museum or art gallery. However, I have known three other threads of meaning.
A misbehaving child is sometimes told to stop making "an exhibition" of herself or himself. Another less desirable use of the word is one I first heard in Egypt some sixty years ago. As male visitors would go ashore in Alexandria or Port Said, they would be accosted by touts, encouraging them to watch an "exhibition", often abbreviated to "exhibeesh". I always resisted the temptation, but I was reliably informed that these were displays of sexual athleticism, sometimes involving donkeys or other creatures, usually relating to a human subject.
It is mainly about another form of "exhibition", that I write today, a usage I have not heard in the United States, but of which I was reminded in a recent communication from my alma mater, Lincoln College, Oxford. It refers to a sort of minor scholarship, although in the UK we used to hear of major and minor scholarships also.
In Britain, a "prep school" refers to a private school for children from about the age of eight until they move on to a so-called "public school" (which is, of course, a private school as opposed to the American version, which one attends if not enrolled in a private school.) Confusing? You bet.
In my day, prep schools and public schools were for boys or girls, but not "co-ed". This has long ago changed in the UK. Other changes may make my comments here rather dated: for example, in my time, prep school boys did not take the notorious "11+" exam which, before the advent of Comprehensive schools as the norm for all children, was used to separate "the sheep" destined for secondary modern schools from "the goats" who were admitted to "grant aided" grammar schools.
In my time, virtually all public schools required those seeking admission to take the Common Entrance (C.E.) exam. There were a few holdouts: in my day, Winchester required applicants to take that school's own examinations, heavily laden with the classics. Whereas I understood the appeal of learning to read and understand history and poetry in Latin and Greek, I felt that the art of turning English prose into Latin - or, God forbid, Greek - verse reflected the sadism of our mentors.
It is possible that some schools offered scholarships or exhibitions to gifted students based on the results of their C.E. papers, but the norm was to require applicants for these honors to take separate scholarship exams. The best results earned a scholarship; others with not-quite-so-good results were awarded an "exhibition".
On the other hand, I have recently also encountered further abuse of the term "scholarship" for those unable to pay their way to a church camp, etc. I am still firmly of the opinion that there are more appropriate ways of describing monetary relief to those who cannot afford to pay the full charges. I don't see the word "bursary" these days, but sometimes I do see reference to "grants" or "financial assistance".