Poetry is the earliest form of literature, and dramatic poetry was probably the first form of poetry. It's fair to say that dramatic poetry, also called dramatic verse, is an ancient art form. Dramatic poetry is meant to be recited or enacted; the "dramatic" label does not necessarily imply that the work is dark. More correctly, dramatic implies theatrical. Shakespeare, for example, wrote all of his plays in dramatic verse.
Dramatic poetry is narrative -- it tells a story -- spoken from the point of view of a persona, a speaker who is a character rather than the author. Often, dramatic poetry has multiple characters. They often speak mostly in rhymed lines, in blank verse or in a combination of the two. Blank verse refers to unrhymed lines of 10 syllables long with every other syllable stressed. Shakespeare wrote his plays in blank verse.
The monologue may have multiple characters but only one speaker. That speaker may or may not be reliable. The reader has to keep in mind that the speaker is telling a story from his point of view only. Were another character to tell the story, the reader would get another point of view. For example, one character, the Duke, tells the story in Robert Browning's "My Last Duchess." He believes that the Duchess gave herself freely to other men, but without the point of view of another more objective speaker, you can't really know whether that is true.
Some dramatic verse is comedy. It may be comedy in the humorous sense that readers think of today, or it may be comedy in the classical sense, in that it ends happily in spite of the sometimes very serious trouble that unfolds throughout the story. Shakespeare's "The Tempest" is a comedy, even though it's not funny, because it ends happily. "As You Like It," another of Shakespeare's comedies, has a happy ending, and it is also funny.
A tragedy is any story that ends unhappily. The verse dramas "Hamlet," "Romeo and Juliet" and "The Crucible" are examples of tragedies. Verse dramas always take place in the present. For example, you read "My Last Duchess" as if you were standing there listening to the Duke speak. Likewise, you watch the events of a Shakespearian tragedy unfold as if they were happening right now.