A dialogue is a conversation between two people, but a monologue is just one person talking. ("Mono" means "one). But "Prufrock" is a "dramatic" monologue because the person talking is a fictional creation, and his intended audience is fictional as well. He is talking to the woman he loves, about whom we know very little except for the stray detail about shawls and hairy arms.
A good dramatic monologue gradually reveals more and more about the person speaking, without them intending to reveal so much. At the beginning Prufrock is just a slightly creepy guy who wants to take a walk. As the poem goes on, we learn about his personal appearance, his love of food and fashion, and his desire to be a pair of crab claws. The impression we get about him is exactly the opposite of the one he wants to give. He wants us to think that he’s a decision-maker, a "decider," if you will, who dresses well and seizes opportunities when they come. But he’s just a big fraud. He never decides anything, and when he misses his big opportunity, he tries to pretend it’s no biggie.
So, the overarching form is the dramatic monologue, but if you look closer at the poem, you’ll find that Eliot is experimenting with all kinds of forms and meters. For example, there are a lot of rhyming couplets, like the first two lines, and the famous verse about the women and Michelangelo.
We think that Eliot is making fun of Prufrock by using this old-fashioned form. The rhyming couplets are sometimes called "heroic" couplets, but our title character is anything but heroic. The rhymes also have a singsong quality that makes them seem childish. He rhymes "is it" with "visit"? Come on. But this is Prufrock’s song, and Eliot is just pulling the strings to make him look bad – quite masterfully, we might add.
Other lines don’t rhyme and sound more like free verse, which has no regular meter. Occasionally we’ll get a couple of lines of blank verse, which have no rhyme but a regular meter, usually iambic pentameter, where an unstressed syllable is followed by an accent. This is the meter that Shakespeare used most often, and Eliot was a huge fan of Shakespeare. Thus, "I SHOULD have BEEN a PAIR of RAG-ged CLAWS."
Shakespeare also used rhyming couplets in iambic pentameter, and, lo and behold, this it the form we get in lines 111-119, which discuss Shakespeare’s Hamlet. Using Shakespeare’s verse to talk about Shakespeare? T.S., you clever man.