There are two separate answers to your question, depending on how you want to structure your business:

1) Foreigners in the U.S. who do not already have a pre-existing business

If you are a foreigner (i.e. non-U.S. citizen or resident green card holder) living in the U.S. on a visa, or living abroad but doing business in the U.S., and you do not already have a company set up in another country, you can form a business in the U.S. However, you cannot form an S Corporation, which has specific restrictions against having foreign shareholders. You may form a C Corporation or LLC.

2) Foreigners in the U.S. who already have a pre-existing business in a foreign country

If you are foreigner who already has a business organized under a foreign country's laws, you may want to register your company as a foreign entity in the state where you'll be conducting business. This involves registering for "foreign qualification" in that state.

You may also want to "check the box" to structure your foreign entity taxes as a pass through for U.S. tax purposes. This is called "check the box" entity classification and is done on form 8832 by the I.R.S.