Notice the rhyming structure of Shakespeare's sonnets. You'll see that each line has a rhyming pair with another line. By naming each line by the rhyme it creates, the structure is typically annotated this way: A, B, A, B, C, D, C, D, E, F, E, F, G, G. This means the first and third lines rhyme, the second and fourth lines rhyme, etc. Though Sir Philip Sidney began this style in the 16th century, Shakespeare made the form famous. The poems also follow the iambic pentameter form, as most of Shakespeare's writing does. This means the rhythm of the poems contain 10 beats, beginning with an unstressed beat followed by a stressed beat, etc.
Note that a poetic ballad is different than the modern concept of a sung ballad. The poetic ballad tells a folk story or legend, and often has a chorus or refrain much like a song. A famous ballad remains "The Rime of the Ancient Mariner" by Samuel Taylor Coleridge, which tells the story of a mariner who shoots an albatross while on a voyage and suffers dire consequences. The poem serves as a morality tale, and presents the ballad form very clearly.
Understand that simply because a poem has no rhyme or metric requirement, doesn't mean it is without form. Many poets, like T. S. Eliot, felt more limited when they didn't have a set structure with which to work. Free verse takes many forms, and can seem more like an excerpt from a book than a poem; yet, at closer examination, these poems are filled with imagery, alliteration, assonance, and conceit (or metaphors). There's always a palpable rhythm to these poems as well. One great example is Walt Whitman's "After the Sea-Ship."
Pay attention to the rigid structure of the sestina when you read one. A great example of this type of poem is actually called "Sestina," written by Elizabeth Bishop. The structure contains 7 parts. The first 6 parts are 6 lines each, and can be any length and do not rhyme; however, each word that ends each line must repeat in a specific structure. By numbering the last word for each line in the first segment 123456, it's clear in the second segment the 6th word has moved to the first position, making the second segment number 612345. This continues until the seventh segment, which contains only three lines. These three lines must contain all of the numbered words within them. The poem, in total, is 39 lines long and must adhere to this structure to be called a sestina.
Don't underestimate the power of the haiku. This Japanese form originated in the 17th century and continues in the present day. The traditional form consists of a mere three lines, with syllables numbering 5, 7 and 5, in that order. These poems concentrate mostly on nature, metaphors, and the study of the sacred in ordinary things. Although they might be the shortest poems, their beauty is startling in its simplicity.