Please tell me the rule when using a/an. I know that "an" should be used before a vowel word, but "A university" is it correct? Why?
Not "word", but sound. Use an before a word that has a vowel sound. Like this,
The words USA and university start with the letter <u>, and that letter is pronounced with a consonant sound, transcribed as either [y] or [j] depending on your dictionary:
USA: [yu]SA, [ju]SA
university: [yu]niversity, [ju]niversity
a USA citizen
a university class