"A Glossary of Literary Terms" by M.H. Abrams defines a dramatic monologue by three characteristics. It must come from a single character (not the writer himself) and constitute the entire poem; it must be directed at an existing listener, whether present or inferred; and it must reveal some aspect of the character to the listening audience.
Dramatic monologues date back to the earliest forms of classical writing in Greece, but they reached their heyday in the late Romantic period of Victorian England. Works such as Alfred Tennyson's "Ulysses," Robert Browning's "My Last Duchess" and Matthew Arnold's "Dover Beach" constitute the pinnacle of the dramatic monologue, though the practice continued into the modern and postmodern era with works from T.S. Eliot and Sylvia Plath.
A dramatic monologue often takes on an assertive or argumentative tone, intended to state a specific case for the implied listener. This is in keeping with the Romantic tradition, which often sought to create a balance between the reader's sympathies and his condemnation. It becomes "poetry of experience," allowing the reader access to a range of emotions seemingly as they happen in the heart of the character.
The vagaries of opinion and emotion portrayed in dramatic monologues are intended to shine insight into the character. For instance, in Tennyson's "Ulysses," the titular Greek hero is reminiscing on his adventures in his old age and wishing he could return to sea again. Because the opinions are subjective, they illuminate the emotional state of the character. More importantly, they leave exact interpretation open to the reader or listener--involving him in the creative process just as much as the writer.
It's easy to confuse a dramatic monologue with a soliloquy, which is used in the theater and which has similar characteristics . Soliloquies consist of characters speaking alone, with the words representing their private thoughts. (Examples include Hamlet's famous "to be or not to be" speech). Though only a single character delivers a dramatic monologue, he speaks to a listener, who may be inferred, but definitely exists.