If you're paid as a contractor on a 1099-MISC then the IRS views you as having your own business, even though you may not think of it that way. It's not the same as the business that's issuing you the 1099-MISC; it's your own business. So when TurboTax asks you about the owner of the business, it means you.
You'll need to set up this business in TurboTax Self-Employed, and the software will walk you thru the process step by step, asking you various questions and allowing you to enter both your 1099 income as well as any expenses you incurred. One of those questions will be whether the business has a name & address different from your own. You can just answer "no" to that, since the IRS doesn't require any special name or office address.
In the end, TurboTax will prepare a Schedule C which shows your business income and expenses. That schedule will become part of your tax return.
