You will almost never hear the word 'posh' used within the U.S., except in reference to Victoria Beckham in a bar trivia question about her nickname in the Spice Girls. But I will assume you're referring to a more distinctive style and dialect by people of the upper-class.
First, there is a language difference. Speaking in more formal and proper English would evoke the image of an upper-class and learned person. This includes not only proper grammar, but minimizing slang and informal words. For example, compare: "I ain't gunna do nuttin." vs "I will do nothing."
Second, there is a difference in the accent and manner of speaking. Specifically, I think what you're most looking for is Received Pronunciation (sample), which is/was the standard British English pronunciation among on the educated elite. This is also the pronunciation most similar to the aristocratic Boston accent.