When do I use 'a', 'an' and 'the' before words starting with 'h'?

The rule depends on sound, not spelling. Use 'a' before a consonant sound and 'an' before a vowel sound. In 'hour' and 'honest', the 'h' is silent, so the word begins with a vowel sound and takes 'an' (an hour, an honest man). In 'hotel', 'hospital' and 'horse', the 'h' is clearly pronounced, so it begins with a consonant sound and takes 'a' (a hotel, a horse). 'The' does not change for sound, but its pronunciation does: 'the' sounds like 'thee' before vowel sounds and 'thuh' before consonant sounds. Quick test: say the word aloud and listen to the first sound, then pick the article.