First, let me note that some people use “an” as the indefinite article form before “historic”, “horrific”, “hotel” and a couple more words beginning with an “H”. However, virtually all speakers pronounce the “H” at the beginning when the word is not preceded by the indefinite article.
Since these words do not begin with a vowel, the rule prescribing the “an” form does not apply to them. In the past, both “an hotel” and “a hotel” were commonly used because English words of French origin beginning with an “H” (such as “hotel”) used to be pronounced without it (so “hotel” would be pronounced just “otel”). Nonetheless, current usage is inclined towards using “a”, not “an”, and always pronouncing the “H”. Not following the rule is perceived as uneducated by some, so you may want to follow it just to be on the safe side.
Nevertheless, there are a few words with a completely silent “H” (“H” which is not pronounced at all in any form of the word). The following list is meant primarily for English learners, so it also contains notes about other common pronunciation mistakes made by learners:
hour /ˈaʊə/ (aau-ə) UK, /ˈaʊɚ/ (aau-rr) US (both the same as “our”); the “h” at the beginning is silent, as it should be also in the name of the letter H /eɪdʒ/ (eydzh). Some native speakers started to pronounce H as “heydzh” lately, but such pronunciation is regarded as incorrect by many.
honour /ˈɒnər/ (on-ə) UK, honor /ˈɑːnɚ/ (aan-r) US; some learners mispronounce this word as if it had /ʌ/ at the beginning (as in “onion” /ˈʌnjon/).
honest /ˈɒnɪst/ (on-ist) UK, /ˈɑːnɪst/ (aan-ist) US; “hon” is pronounced exactly the same as in the previous word.
Hannah /ˈhɑnə/ (haa-nə) UK, /ˈhænə/ (hæ-nə) US; on the other hand, it is the final “h” that is silent in this name, not the first one. The same is true for all words of Hebrew origin ending with “ah”, e.g. Bar Mitzvah.
heir /ɛə/ UK, /ɛr/ US; a person who inherits something from someone else. It comes from Old French, so the “H” remains silent; it sounds exactly the same as “air” and “ere” (meaning “before long”).
exhausted /ɪɡˈzɔːstɪd/ (ig-zaw-stid); notice also that “x” at the beginning is pronounced as “gz”, not as “ks”.
Thai /taɪ/; “th” in English is usually pronounced as /θ/ (as in “think”) or /ð/ (as in “the”), but in “Thai”, is is pronounced just as “t”. The same applies to Thailand too, of course.
vehicle /ˈviːəkl/ (vee-ə-kl); some speakers of American English pronounce the “h”, but the vast majority keep the “h” silent and consider the pronunciation with an “h” unnatural.
ghost /ɡəʊst/ (gəust) UK, /ɡoʊst/ (gohst) US; to end our discussion about the silent “H”, notice that it is also silent in “ghost”. This is in fact the case with all words beginning with “gh” such as ghetto /ˈgɛtəʊ/ (ge-təu) UK, /ˈgɛtoʊ/ (ge-toh) US, ghastly /ˈɡɑːstli/ (gaast-lee) UK, /ˈɡæstli/ (gæst-lee) US, and ghee /ɡiː/ (g-ee).
Note: The word “herb” and words derived from it (such as “herbal”) are usually pronounced with an H at the beginning in British and Australian English, while it usually remains silent in American and Canadian English.