Dramatic monologue is spoken verse that allows a reader or a spectator (if it is a play) to gain insight into the speaker’s innermost feelings or thoughts. It is not the same thing as a soliloquy, however, because a dramatic monologue has an audience whereas a soliloquy does not. In a soliloquy, a character is speaking to himself or thinks he is speaking to himself, but in a dramatic monologue, people are listening.
Some famous dramatic monologues are My Last Duchess, by Robert Browning, Ulysses, by Tennyson, The Lovesong of J. Alfred Prufrock by T. S. Eliot, the ghost of Hamlet’s father speaking to Hamlet in the beginning of the play, Lady’s Macbeth’s monologue to Macbeth when she is trying to convince him to kill Duncan (it begins, “Was the hope drunk wherein you dress'd yourself?), etc.
I would just add something to the already given answer on the dramatic monologue.
Dramatic monologue has a silent listener whose presence has to be recorded in some way or the other in the poem and this recording is crucial. For example, Shakespeare's Sonnet 116 where there is an imaginary courtroom argument between a supporter of eternal love and its critic. The "O no!" in the fifth line is addressed directly to the critic who is the silent listener in this poem.
The contribution of dramatic monologue to the modern poetic idiom is its deep psychological import. In modern poetry from Eliot to Heaney this silent referencing of an implicitly present second person has shaped the idea of a schizoid split or an inbuilt idea of the reader or audience, as theoretically developed by the Reception Theorists.