Utopia essentially invented a new genre: Utopian, or Dystopian, Literature.
Have an inkling of what that might mean? It's when you use elements of fantasy and science-fiction to describe a place that doesn't exist (usually it's far, far away or in the distant future) as way to think about issues of government, society, justice, etc. So, maybe you're getting how the philosophy fits in here, too? Although More's Utopia coined the term for this literary style, Plato's Republic is arguably the first Utopian/Dystopian work of literature, so it definitely has some pretty serious roots in the world of philosophy.
Wait, but what's dystopian? Dystopian is a term used to specifically describe fantastic worlds that are clearly meant to be horrible: Do Not Live This Way! Unlike Utopian literature, which suggests that the world being described might be better than our world, dystopian literature is definitely not. Anyone heard of The Hunger Games? A perfect dystopian example.
But once you read Utopia (and, for that matter, go read the Republic too!) you'll probably start to understand that this distinction isn't so clear. Is Utopia really better? Many characters in the book don't seem to agree. Maybe all utopias are also dystopias...? We're going to leave you with that deep thought.