Dramatic poetry is drama, written in verse form, that is either sung or could be.
Sub-categories of dramatic poetry include comedies and tragedies. Ballads have been considered dramatic poetry as well.
Although today, Shakespeare's plays are not really considered dramatic poetry so much as they are considered dramas, most of his plays are written in (or adhere closely to) iambic pentameter.
Probably Sophocles is a more common name in dramatic poetry. Oeidpus and Antigone are two of his most famous plays that were originally written with a very rigid structure and could have been considered dramatic poetry.
Generally speaking, it is thought that dramatic poetry originated with the very first dramas, as it was a way to write long lines that were easily memorized. Since then, the term "dramatic poetry" is rarely used, as we've branched our literature into either "drama" or "poetry." I think it goes without say though, that most classic drama is written in verse form.