In addition to Som’s correct answer, it should be noted that the H sound is usually dropped by all speakers except if it is immediately before a stressed vowel or at the beginning of a phrase. (Also: English speakers never pronounce the H as strongly as the similar sound in Russian.)

It's <em>eɪtʃ </em>(eich). The first and second instances look like a Cockney accent. They drop their Hs in words but pronounce the letter itself like hheich. That is just a dialectic accent, not the technically correct mainstream form. The last two instances are something that can happen naturally in fluent rapid speech to anyone, without any conscious intention on the part of the speaker or any distinct accent like Cockney. Hotel, historian and hysteric all have a sounded H, but it bifurcates for herb. The British say it with a sounded H while Americans say erb.

1: 'Aitch' or 'Haitch'? 'Aitch' is the standard, 'correct' way to refer to 'H', but in fact millions of native speakers call this letter 'haitch'. By the way, this has nothing to do with accents (Cockney or otherwise). 'Haitch' is not a mispronunciation of 'aitch': it is a separate, non-standard variant. You'll come across this variant in speakers of all accents, including some RP speakers who would never dream of dropping an initial stressed 'h' or putting one where it doesn't belong. It's not mainstream, but learners do need to be aware of it, particularly in Britain and Ireland.

Interestingly, aitch/haitch can serve as a religious shibboleth: in parts of the UK with large communities of Roman Catholics of Irish heritage such as Liverpool and Glasgow, you'll apparently find that Catholics say 'haitch' while Protestants are more likely to say 'aitch'.

2: The only words where the initial 'h' is always silent are 'hour' and 'honour', and related words (hourly, honesty and so on). That's just one very simple rule which you need to remember. Not too difficult, surely?

[Minor points: 'Hotel' and 'historic' (and related words) used to have a silent 'h', but this is very outdated. You'll occasionally come across phrases such as 'an historic moment', but that's very rare nowadays. Don't worry about it. 'Herb' pronounced 'erb' is a feature of only AmE - both pronunciations are correct].

3. What happens in natural speech - elision - is a different issue entirely. As Phil says: "<em>the H sound is usually dropped by all</em><em> </em><em>speakers except if it is immediately before a stressed vowel or at the beginning of a phrase</em>". We do indeed say 'tell'er' rather than 'tell her' and 'we'd'uv come' rather than 'we'd have come'. This is perfectly normal. Our speech would sound very strange and unnatural if we pronounced every 'h', just as it would sound unnatural if we pronounced every syllable with equal stress.

Som: Nice example of Hs that are always pronounced (i.e. not dropped) in standard accents.

Phil, you mean <em>Hhell hhath no fire like hhatred </em>is incorrect?