As a US citizen, you must declare your worldwide income to the IRS no matter where in the world you live.
When you live outside the US for a period of 12 months or more, you are eligible for the overseas earned income exclusion, which means that you can exclude up to about $91,000 of salary earned while resident overseas. (The idea being that you are paying taxes locally to your country of residence - though even if you're living in a non-tax country, you can take the exclusion.)
For amounts over and above the exclusion that you receive as salary or any amount of "unearned income" (i.e. investment income, interest, rents, etc.) you can take a foreign tax credit for taxes paid to the local government.
For your US taxes, everything is calculated exactly the same way it has always been done, with the exception of the earned income exclusion and the foreign tax credit. For details about how these work, download publication 54 from the IRS website.