Hi! Hello! How do you do? H is a happy, hearty, hospitable letter. It is also hale and healthy (the aspirate sound demonstrating that the speaker is definitely breathing) and given to bouts of hilarity - even hysteria (ha ha, hee hee, ho ho).
Helpful to the last, h appears in all the main consonant digraphs (sh, ch, th, Greek ph) and two common letter-strings (igh, ough). It also appears in wh, but whether you consider that a digraph or a silent h depends on your pronunciation.
Interestingly, the wh words (including the five "question" words, what, when, where, why, and which) originally had h upfront - the Old English hwt, hwanne, hwr, hwy, hwilc. The reversal presumably happened because the w sound was dominant.
H is silent in some words borrowed from other languages, such as the Slavonic gherkin, Italian ghetto, Hindi ghee, and Arabic ghoul. The h in ghost, though, is there courtesy of William Caxton, who inserted it in Old English gast to analogise with the Flemish gheest.
In a few words the initial h is silent, such as hour, honest and, if you are American or very old, herb and hotel. Mind you, for many children in my classes, the initial h was always silent, as celebrated in Anon's little rhyme: Arry ad an awk in an andbag, And the awk made an orrible row.
Arry it the awk wiv an eavy ard ammer Arry ain't got an awk now.