Dramatic poetry is poetry written specifically for the theater. This type of poetry can often be lyrical in nature, such as when a character in a play gives a dramatic monologue.
Dramatic monologues are a very common form of dramatic poetry. In a dramatic monologue, the speaker addresses a kind of imaginary listener, so they are not explicitly speaking to the reader. This type of poetry is well fit for the dramatic context because characters in a theater production may often have small speeches where they address an imagined listener and, in many cases, the audience fills in that role. This type of poetry can have a song-like quality to it, using lyrical elements or even employing a more narrative structure. In this way, dramatic monologue, and dramatic poetry in general, isn't limited by its associations with drama and the theater. Instead, dramatic poetry appears in numerous varying contexts, utilizing many different poetic devices and variations within the context of the dramatic medium.
Some classic examples of dramatic monologues, in particular, are "My Last Duchess" by Robert Browning, "The Love Song of J. Alfred Prufrock" by T.S. Eliot and "Killing Floor" by Ai. Some examples of dramatic poetry could be "After Hearing a Waltz by Bartok" by Amy Lowell, "The Battle of the Bards" by Theocritus, "A Servant to Servants" by Robert Frost or "The Lady and the Painter" by Robert Browning.