Verse drama is any drama written as verse to be spoken; another possible general term is poetic drama. For a very long period verse drama was the dominant form of drama in Europe (and was also important in non-European cultures). Greek tragedy and Racine's plays are written in verse, as is almost all of Shakespeare's drama, and Goethe's Faust.
Verse drama is particularly associated with the seriousness of tragedy, providing an artistic reason to write in this form, as well as the practical one that verse lines are easier for the actors to memorize exactly. In the second half of the twentieth century verse drama fell almost completely out of fashion with dramatists writing in English (the plays of Christopher Fry and T. S. Eliot being possibly the end of a long tradition).
Dramatic poetry is any poetry that uses the discourse of the characters involved to tell a story or portray a situation.
The major types of dramatic poetry are those already discussed, to be found in plays written for the theatre, and libretti. There are further dramatic verse forms: these include dramatic monologues, such as those written by Robert Browning and Alfred Tennyson.