"University" doesn't really begin with a vowel. It begins with a "y" sound (like "you") which is a consonant. It's sound, not spelling, that's important. This is why we don't say or write "an university".

Also, in many accents of English, initial "h"s are dropped. Which means "house" does begin with a vowel for those people. They therefore say "thee 'ouse" and "an 'ouse", not the standard pronunciations "thuh house" and "a house" (please excuse the eye-dialect).

However, if we substitute better examples, such as "apple" and "banana", your English teacher is correct (at least for British English – the Americans who posted above seem to disagree). And this is definitely not one of those "zombie rules" which only English teachers believe in and bear no relation to the way people actually talk in the 21st century, like "split infinitives". This is what (British) English speakers actually do.

(Incidentally, I first became aware of this distinction when singing in a choir – when the next word after "the" was over the page, and I was expecting it to be "earth", I would sing "thee", and then get caught out when it turned out to be "world" instead – "thee earth" is right but "thee world" just sounds wrong.)