A "monologue" is like a long speech by a single actor/character in a play (or in a book / movie / TV show). The person is talking to themselves (and the audience), but not to the other actors/characters.
"Dramatic" can mean "theatrical" or "exaggerated"
A "dramatic monologue" (or a "persona poem") is basically a poem presented in the style of a monologue where a character (in a play / book / movie etc.) is speaking the words of the poem aloud (to the audience) but they are not the words of the character (the poem is just being delivered to the audience through the character). The poem may reveal information about the character, but from a third-person perspective.
It's quite difficult to explain. There are some examples online, but it's not that easy to understand.
@kia03 isn't that just a regular monologue though? As far as I know "dramatic monologue" has a very specific meaning and it involves a poet (third-party) speaking through the voice of a fictional character who is delivering the poem to an audience in the form of a monologue.
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