Haiku (hokku, haikai): a Japanese form that Western poets tend to render as 3 lines of 5, 7, and 5 syllables. 

the sea darkens -- the voices of the wild ducks are faintly white. Matsuo Basho, 1644-1694

Terza rima: a continuous interlocking rhyme scheme (ababcbcdc...) most famously used in Dante's Commedia. Many poets render it in tercets (aba bcb cdc...).

O WILD West Wind, thou breath of Autumn's being— Thou from whose unseen presence the leaves dead Are driven, like ghosts from an enchanter fleeing, [...] Shelley, "Ode to the West Wind"

Blues: tercets in which the second line is a refrain of the first (giving the singer time to improvise the third). 

Villanelle, which poets writing in English render as five tercets and a quatrain, originated in a French song form with repeated refrains. The English version is governed by a strict pattern of refrain and rhyme generated out of the first tercet, so that each subsequent stanza repeats a line (or two, in the case of the final quatrain) from the first tercet, and the entire poem is restricted to the two rhyme-sounds introduced in the first tercet. (See The Poetry Foundation for some famous examples.) 

Ballad stanza: most commonly, alternating iambic tetrameter and trimeter (4, 3, 4, 3 feet), rhyming abxb, although there are numerous variations. Aside from the line breaks, a ballad stanza is metrically equivalent to a fourteener couplet. Note a poem written in ballad stanzas is NOT necessarily a ballad, and vice versa.

 A SLUMBER did my spirit seal; I had no human fears: She seem'd a thing that could not feel The touch of earthly years. [...] Wordsworth, "A Slumber Did My Spirit Seal"

Heroic Quatrain: iambic pentameter rhyming abab (i.e., alternating heroic couplets).

THE CURFEW tolls the knell of parting day, The lowing herd winds slowly o’er the lea, The ploughman homeward plods his weary way, And leaves the world to darkness and to me. Thomas Gray, "Elegy Written in a Country Churchyard"

"In Memoriam" stanza: iambic tetrameter quatrains rhyming abba, named after Tennyson's long elegy for Arthur Hallam. 

THOUGH beauty be the mark of praise, And yours of whom I sing be such As not the world can praise too much, Yet 'tis your Virtue now I raise.. Ben Jonson, "An Elegy"

Common or hymn measure alternates tetrameter and trimeter (like a ballad stanza) and rhymes abab.  

The pantoum (a variant on the Malaysian, is a series of quatrains rhyming abab, in which the even-numbered lines (2 and 4) of each quatrain are repeated as the odd-numbered lines (1 and 3) of the next quatrain.

Limerick: an anapestic trimeter triplet surrounding an anapestic dimeter couplet---i.e. the number of feet per line is 3, 3, 2, 2, 3, rhyming aabba. Limericks often use feminine rhyme, adding to the rollicking comedic effect.

The Complete Nonsense of Edward Lear (an illustrated collection of limericks)
Wendy Cope, "Waste Land Limericks"

I Wish that my Room had a Floor! I don't so Much Care for a Door, But this Crawling Around Without Touching the Ground Is Getting to be Quite a Bore! Gelett Burgess, "The Floorless Room"

Sapphics: English poets have adapted this quantitative form in many different ways, although the shortened final line is a reliable indicator. The traditional English Sapphic consists of lines of 11, 11, 11, and 5 syllables in falling meters (dactyls and troches), and need not rhyme.  

How happy he, who free from care The rage of courts, and noise of towns; Contented breaths his native air, In his own grounds. Alexander Pope, "Solitude: An Ode"

Venus and Adonis stanza: iambic pentameter lines rhymed ababcc, named after Shakespeare's "Venus and Adonis", which uses this form.

EVEN as the sun with purple-colour’d face Had ta’en his last leave of the weeping morn, Rose-cheek’d Adonis hied him to the chase; Hunting he lov’d, but love he laugh’d to scorn; Sick-thoughted Venus makes amain unto him, And like a bold-fac’d suitor ’gins to woo him. William Shakespeare, "Venus and Adonis"

Tail Rhyme: A medieval verse form in which a "tail" line rhymes to make a couplet embracing two other couplets, e.g. aabccb. Tail rhyme is often, but not always, in sestets.

THREE years she grew in sun and shower, Then Nature said, "A lovelier flower On earth was never sown; This Child I to myself will take; She shall be mine, and I will make A Lady of my own. William Wordsworth, "Three Years She Grew in Sun and Shower"

Sestina: six sestets and a tercet, whose pattern is determined by end-word repetition rather than rhyme or meter. Each sestet repeats, in a spiral pattern (6, 1, 5, 2, 4, 3), the end-words of the previous sestet; the final tercet orders the repeated words according to the same spiral pattern but uses two per line, one in the middle and one at the end.

Rhyme royal: seven pentameter lines rhyming ababbcc

The double sorwe of Troilus to tellen, That was the king Priamus sone of Troye, In lovinge, how his aventures fellen Fro wo to wele, and after out of Ioye, My purpos is, er that I parte fro ye. Thesiphone, thou help me for tendyte Thise woful vers, that wepen as I wryte! Geoffrey Chaucer, Troilus and Criseyde

Ottava rima : pentameter lines rhyming abababcc.

I WANT a hero : an uncommon want, When every year and month sends forth a new one, Till, after cloying the gazettes with cant, The age discovers he is not the true one ; Of such as these I should not care to vaunt, I'll therefore take our ancient friend, DON JUAN, We all have seen him in the pantomime Sent to the devil somewhat ere his time. George Gordon Byron, Don Juan

Spenserian stanza: 8 lines of iambic pentameter followed by an alexandrine, rhyming ababbcbcc.

A Gentle Knight was pricking on the plaine, Ycladd in mightie armes and siluer shielde, Wherein old dints of deepe woundes did remaine, The cruell markes of many' a bloody fielde; Yet armes till that time did he neuer wield: His angry steede did chide his foming bitt, As much disdayning to the curbe to yield: Full iolly knight he seemd, and faire did sitt, As one for knightly giusts and fierce encounters fitt. Edmund Spenser, The Faerie Queene

A sonnet is a fourteen-line poem, usually divided into an octet and sestet (Petrarchan) or three quatrains and a couplet (Elizabethan and Spenserian). A sonnet sequence is a series of sonnets (usually divided by white space, often separately numbered) that share a related theme.