Even though there may or may not be any W-2 or other tax form received by you and field by a foreign company this is still either wages and salary or self employment income that has to be reported by a citizen or resident of the United States.
If you qualify it may be possible to exclude foreign earned income (though it is still reported and then excluded) from taxation by the US.
"To claim the foreign earned income exclusion, the foreign housing exclusion, or the foreign housing deduction, you must have foreign earned income, your tax home must be in a foreign country, and you must be one of the following:
- A U.S. citizen who is a bona fide resident of a foreign country or countries for an uninterrupted period that includes an entire tax year,
- A U.S. resident alien who is a citizen or national of a country with which the United States has an income tax treaty in effect and who is a bona fide resident of a foreign country or countries for an uninterrupted period that includes an entire tax year, or
- A U.S. citizen or a U.S. resident alien who is physically present in a foreign country or countries for at least 330 full days during any period of 12 consecutive months."
Even if you can not qualify for foreign earned income exclusion (FEIE) if you pay tax on the same income to both the foreign government and the US then you get to claim a credit for the taxes paid to the foreign government on Form 1116.
Links and details are at http://www.irs.gov/taxtopics/tc856.html
For more complete details see http://www.irs.gov/publications/p54/index.html
If you are an employee the earned income is included on line 7 of Form 1040 and if you are self employed it will be reported on Schedule C.
You will also claim an exclusion on Form 2555 or a credit for foreign taxes on Form 1116. See links above for more information.
Please ask if you need more discussion or clarification.