A dramatic monologue is a poetic form in which the poet takes on a persona as the speaker of the poem. This speaker addresses a silent listener called the auditor (this is what differentiates a dramatic monologue from a soliloquy).
Dramatic monologues usually occur during a critical moment of the speaker's life, and there is usually a dramatic circumstance behind the poem-- however, the point of the poem is not as much to tell the story as to explore the character of the speaker. Typically, these monologues have a certain aspect of dramatic irony, in which the reader can infer something from the speaker's words that the speaker themself does not know or does not intend to reveal. This can be a different perspective on the story, where the reader knows it didn't happen quite like the speaker tells it, but most often it is an aspect of the speaker's personality.
Robert Browning is credited with perfecting the form of the dramatic monologue with poems such as Porphyria's Lover and My Last Duchess. Other good examples of dramatic monologues include T.S. Eliot's The Love Song of J. Alfred Prufrock and Tennyson's Ulysses.