H-dropping, the technical term for the more common [bad] habit of "dropping your aitches", refers to the tendency of many speakers to omit the initial "h" in words like her ('er), hair ('air) or half ('alf).
There are, in fact several instances of the written "h" being correctly omitted in the spoken word. Thus, we have hour, honest, honour, heir, etc. Likewise, in the 19th Century, it was normal to pronounce hospital, hotel and herb without the initial "h". Michael Quinion, in his excellent World Wide Words, points out that until the sixteenth century the word herb was usually spelled erb, the "h" being added later: the British people only began to say the first letter as a result of spelling pronunciation, during the nineteenth century.
Aitch or haitch?
According to Jo Kim at the BBC Pronunciation Unit, there is now a tendency for younger people to [possibly] hyper-correct and insert aitches where they don't exist. On the other hand, while aitch /eɪtʃ/ is regarded as the standard pronunciation of the letter "h" in British English, haitch /heɪtʃ/ is a standard pronunciation in Irish English.