Why is it "an hour" but "a university"?
Because the choice between a and an depends on the sound a word begins with, not the letter.
The actual rule: use an before a word starting with a vowel sound, and a before a word starting with a consonant sound.
Why "an hour". The h in hour is silent, so the word begins with the vowel sound /aʊ/. It is pronounced as though it started with ow, so it takes an. The same applies to an honour, an honest mistake, an heir.
Why "a university". Despite starting with the letter u, the word begins with the consonant sound /j/ — a y sound, as in yoo-niversity. It therefore takes a. The same applies to a European, a unicorn, a one-off (which starts with a w sound), and a useful tool.
Compare with an umbrella, an uncle, an understanding — these genuinely begin with a vowel sound.
The rule extends to abbreviations, and this catches people:
An MP, because M is pronounced em.
An NHS trust, N being en.
An FBI agent, F being ef.
A UN resolution, because U is pronounced yoo.
A US citizen, same reason.
The test is always how you say it, not how it looks.
Where usage varies legitimately. Words beginning with a pronounced h in an unstressed first syllable — hotel, historic, horrific — were traditionally given an in some British usage, from a period when the h was more often dropped. "An historic occasion" survives as a stylistic choice and is not an error, though "a historic" is now more common and is what most style guides recommend. Note that an only makes sense if the h is unpronounced; "an historic" with a clearly sounded h is inconsistent.
Herb is an herb in American English, where the h is silent.