The sonnet is the most standard sort of fixed-form poem; also, there are many variations on the basic fourteen-line pattern.
Beyond that fourteen-line length, what makes a sonnet a sonnet? Mostly, it’s a matter of stanzaic form. The two chief kinds are the Italian or Petrarchan and the English or Shakespearean. The Italian looks like this:
So, two quatrains in “envelope” rhyme, using the same rhyme sounds, and three couplets (two-line stanzas) in alternate rhyme with each other, also using the same rhyme sounds (though different from the first set in the quatrains). Usually, though, the Italian sonnet is thought of as an octave (eight-line stanza) followed by a sestet.
Or, three quatrains using alternating rhyme, each pair of sounds different in the three stanzas, ending with a rhyming (or “heroic”) couplet.
Traditionally, sonnets are printed in monolithic, or single-stanza form; the stanzas are not spatially separated. But the energies of the stanzaic differences are quite important. The second- most important formal point, after the fourteen-line length, is the “turn.” Actually, though, this is a tonal or a thematic issue, more than formal. The turn is a shift of some kind roughly between the octave and the sestet in the Italian, and the third quatrain and the couplet in the English. You aim for some kind of modulation in your language, point of view, narrative (if there is one – most sonnets are not strongly narrative, though, although most poems suggest an outer narrative), or emotion. You can see, then, that the Italian allows a lot of room for exploring this second modality, whereas the English allows only two lines – making the second modality, often, a kind of punch line, or a surprise.
There are, as I said, many other forms of the sonnet. There are, for example, hybrids of the English and Italian; and couplet sonnets (seven rhyming, heroic couplets). Again, you should start with the text, then move to the Internet for examples and further explanations. You might choose, by the way, to do these fixed forms as “pastiches” – imitations of other poems, that is, attempts at copying the style (and perhaps content) of the other poem. It is a good way to practice, just as in music and painting.
“Sonnet Central” is a very good Internet resource to start with; look up John Keats, John Milton, Elizabeth Barrett Browning, to start. Here are some others to look at:
James Wright, "Saint Judas" (scroll down a bit)
Edna St. Vincent Millay, "Sonnet VI"
June Jordan, "Sunflower Sonnet Number Two" (In this article, by Adrienne Rich, about the poet’s work; read, or scroll down to find the poem)
Seamus Heaney, "Requiem for the Croppies"
So: write a sonnet! It can be an imitation of one you find, or an experiment of some kind; combine rules from different sonnets, or invent new rules.