Shakespeare's sonnets were greatly admired even before they were published; as with the informal verses of many other poets of the day, they were circulated in manuscript. It might be something like the present day, when the poetry or prose of a writer circulates online, via their blogs or those of their friends, before being picked up by a publisher to be "officially" brought out to the public. It's not even clear whether Shakespeare himself authorized their publication, although the case is better for the sonnets than for the plays. It was, like the early days of the Internet, a wild time for publication, with piracy and bootleg editions rampant.
On a formal level, it is important to distinguish the "Shakespearian" sonnet from its "Petrarchan" precursor (in fact created by Giacomo da Lentini). Both are entirely in iambic pentameter; the Petrarchan form was made up of an "octave" (two quatrains of four lines) rhyming abba/abba, followed by a closing sestet (cde/cde). The octave was supposed to describe some sort of quandry or problem, which the sestet would, to some degree, answer. It was Spenser, in English, who changed the division of the lines from the octet and sestet to three quatrains and a two-line envoi -- here the three quatrains could develop three takes on a single theme or a series, to which the envoi did not need to offer a clear answer. And it was this far more elegant division that Shakespeare took up, following it in every one of his sonnets.
The order of the sonnets is uncertain -- neither published version had Shakespeare's explicit sanction -- but there are two broad sequences that readers then and now have discerned: the first are largely addressed to a male friend or lover, urging himself to outwit the passage of time and preserve his beauty by having offspring; the second series seem addressed to a harsh if not cruel "dark lady," a woman who has scorned the poet's attentions.
But these poems only live because we, every time we read them, make them new -- finding meanings suited to our time and place -- and thus our own understandings matter just as much as all of the above. So pick a sonnet -- any sonnet -- and describe what it means to you, and to whom (an envoi is a letter, or an ambassador) you would send it.