What type of poetry did Edgar Allan Poe write (haikus, free verses, limericks)?

Edgar Allan Poe primarily wrote narrative poems, poems which tell stories in metered verse. While Poe's work wasn't explicitly meant to be recited or sung, he referred to many of his narrative poems as "ballads" for their highly lyrical and dramatic quality.
"Annabel Lee," one of Poe's most famous works (of poetry or otherwise), demonstrates his general style and approach to poetry. The common characteristics of this and many of his other poems are: alternating yet imperfect rhyme schemes, traditional stanza structure, and considerable length. (These are certainly no haikus!)
Consider the opening lines of "Annabel Lee:"
It was many and many a year ago,
In a kingdom by the sea,
That a maiden there lived whom you may know
By the name of Annabel Lee;
And this maiden she lived with no other thought
Than to love and be loved by me.

The stanza structure is traditional, and governed by an alternating rhyme scheme. Generally, the scheme is "ABAB," but Poe does not abide by this scheme rigidly. The poem tells a story about the death of a beautiful girl in a tone of rapture and despair. Poe's emotionally charged subject matter suits the narrative form. The poems harken to love songs or elegies with their rhyme schemes, and are long enough to create compelling narrative arcs with fully developed subjects.