Generally, it’s either « les États-Unis » or « les USA ».
Ironically, we barely use « les EU », apart from formal writing.
Here’s a quick list of the terms we use :

Les États-Unis

By far the most used, you can use it in almost any situation.

Les USA

Less formal, can sometimes be seen as an attempt to ‘brag about one’s mastery of English’, because why say it in English when we have a word for this ?
That’s why it is more used in books or on the radio than in everyday life. It’s relatively common though.

Les E.-U. / EU

Apart from books, I don’t know anyone who uses it…

US

Same here, I mostly see it in books about English, i.e. dictionaries, trip books, and the like.

The States

Unless it’s obvious you’re from there, never ever use it. It is THE word for people who don’t know a thing about English, but still want to pretend.

To sum up, in general, « États-Unis » > « USA » > « EU – US » > « States ».
Voilà !

To complete the good answer of Nathan, it must be noted that the adjective form behaves differently.

Indeed, l’Amérique points to the continent, and les États-Unis, les US or – more colloquial – les states are proper ways to refer to the country. Nevertheless, the corresponding adjective is américain as in “les voitures américaines”. It is expected to rely on context to tease apart “américain” for the country or for the continent, with a huge bias towards the country.

The “proper” form “étatsunien” exists but is extremely rarely used, especially orally.

Another common expression that isn’t mentioned: if you want to say “in the USA”, then you can say “outre-Atlantique”, literally “over the Atlantic”. For example, “Outre-Atlantique, il est de bon ton de manger des hamburgers” = “In the US, it’s good manners to eat hamburgers.” Another example: “Cette voiture nous vient d’outre-Atlantique” = “This car comes from the US.” (An analogous expression, “outre-Manche” = “over the Channel”, refers to GB.)

It cannot be used in all contexts, and it’s a ambiguous as it could refer to anywhere in the Americas, but it’s often used. This is of course an expression centered on Europe, and I doubt someone from Québec would use it, for obvious reasons.