grammaticality - Using "An" and "A" in a sentence

I'm trying to understand this simple concept.

As far as I understood it, back to the days when I was a student, "an" should be used only before vowel words, that is, only before the following words: "a","i","o","u".

Yesterday my sister asked me a question related to this subject and I wasn't sure if my answer was correct because there are so much controversial sources explaining this subject.

The original question she asked me was which one of the following is the correct form to be used in a sentence:

An history

Or

A history

I told her that she should use "a history" only because 'h' isn't a vowel word. Is it correct?

Thank you very much

Answer

Words of one or two syllables beginning with ‘h’ are normally preceded by ‘a’. They include ‘hotel’, ‘hostel’, ‘host’, ‘hearty’, ‘hero’ and ‘hardy’. (Some speakers may say ‘an hotel’, because ‘hotel’ is also a French word, in which the ‘h’ is not aspirated.)

Words of three or more syllables beginning with ‘h’ in which the first syllable is stressed are also preceded by ‘a’. They include ‘history’, ‘herbalist’, ‘heightening’ and ‘helicopter’.

Words of three or more syllables beginning with ‘h’ in which the second syllable is stressed give rise to some uncertainty. They include ‘historian’, ‘historical, ‘hiatus’ and ‘Hibernian’. I and some other speakers of British English precede them with ‘an’, but others don’t.