If you make over US$600 from a client within a tax year they are supposed to send you (and the IRS) a 1099-MISC. Many companies / people do not currently do that, but starting in 2012 they will have to (new tax laws come into effect in 2012). You claim the income on Schedule C.
You *can* set up your business in someplace like Bermuda as a corporation to avoid most taxes, but to do that there is an initial investment, including a lawyer there, to set up and maintain the corporation. And, you should understand the tax codes (at least the basics) in both places so you don't get caught up in a tax avoidance scheme by the US's IRS. It's not hard to do, but unless you're making over US$100K it's not cost effective. Once you hit income of US$250K+ it becomes very effective and worth the cost.