What is the relationship between Art and Politics in Roman art?

This is a very broad question! It's slightly difficult to answer without more context, but here's what I think you're looking for:

When I think of the connection between art and politics in the Roman Empire, the first person I think of is the Emperor Augustus, Rome's first Emperor. He was a master of propaganda, and during his long reign commissioned many gorgeous works of art, some of which live on until this day, that...

This is a very broad question! It's slightly difficult to answer without more context, but here's what I think you're looking for:

When I think of the connection between art and politics in the Roman Empire, the first person I think of is the Emperor Augustus, Rome's first Emperor. He was a master of propaganda, and during his long reign commissioned many gorgeous works of art, some of which live on until this day, that broadcast his political message loud and clear. Here are a few examples:

The Aeneid

The Aeneid was an epic poem written by the poet Virgil, commissioned by Augustus. The work tells the story of Aeneas escaping the fall of Troy and travelling to Italy to found what would later become Rome. The poem is chock full of references to Augustus' greatness. He claimed descent from Aeneas and the goddess Venus, several characters predict his coming greatness, and Aeneas even carries a shield that depicts Augustus' victories in battle, hundreds of years before they even happened!

Don't misunderstand, the Aeneid is a BRILLIANT piece of literature, but it is impossible to divorce it from its political connections.

The Ara Pacis

The Ara Pacis was (and is—it's still there!) a public altar built by Augustus to celebrate the peace he brought to the Roman Empire. It is covered in images that he wanted connected with himself—goddesses representing peace, Rome, fortune, his family protecting the well-being of the state, and all kinds of things he wanted people to think of when they thought of him.

All of Augustus' buildings in some way display the symbols and messages he wanted broadcast to Roman society.

Statues of Livia and Octavia

Augustus was also very big on morality, and he passed a great deal of legislation trying to make Rome more moral in his eyes. A bit part of the propaganda surrounding this movement were his wife, Livia, and his sister, Octavia. In statues they are always portrayed as super virtuous, even occasionally as goddesses like Juno.

As you see, Augustus used art to further his political message. He wasn't the only Emperor to do this, but he did it first and, I would argue, best.