(Rhymed) Couplet
A rhymed couplet consists of two successive lines of verse with similar end-rhymes. Both the grammatical structure and the idea of the rhymed couplet convey a sense of completion.
Three common types of couplets are: (1) a closed couplet, (2) an open couplet, and (3) a heroic couplet. The predominant couplet in English is the heroic couplet. This type of couplet consists of two rhyming lines of iambic pentameter with a pause, usually in the middle of each line.

(Rhymed) Stanza
A rhymed stanza is a division of a poem that consists of two or more lines arranged together as a unit. A stanza's group of lines are usually arranged around a recurring pattern of metrical length and sequence of rhyme. This structure is determined by the number of lines, the dominant meter, and the rhyme scheme.

Narrative Poem
A narrative poem is a non-dramatic poem that tells a story or presents a narrative. A narrative poem may be long or short, complex or simple. Some Types of Narrative Poems: Epics, Ballads, Metrical Romances

Dramatic Poem
A dramatic poem is a poem that employs dramatic form or technique.

Satiric Poem
A satiric poem is a poem that treats its subject with irony and/or ridicule.

Didactic Poem
A didactic poem is a poem that is intended primarily to teach a lesson.

Sonnet
A sonnet is a lyrical poem of fourteen lines, highly arbitrary in form and following one of several rhyme conventions.

Ode
An ode is a ceremonious lyrical poem marked by exhaltation or feeling and style. An ode has varying line length and complex stanzas.

Elegy
An elegy is a sustained and formal poem setting forth the poet's meditiations upon a grave theme (usually death).

Taken from Renée Goodvin