I might add this first part is only useful if you're a US citizen. If you're looking for information about Japanese taxes, scroll to the bottom of this post.
If you are a US citizen living in Japan, you need to pay taxes with both the American and Japanese governments. Since you are living abroad, you get an automatic two-month extension from the regular due date; no additional paperwork is required.
Normal deadline: April 15th
Deadline for US citizens in Japan: June 15th
If you do want a six-month extension, the same rules apply as any other situation; you need to file a Form 6868 with the IRS.
I was particularly interested in the law surrounding US citizens who were in the United States for part of one tax year, and Japan the rest of the time. As long as you stay in Japan for the entirety of the tax year, or 330 consecutive days (regardless of whether or not it's within the tax year, e.g. from June 2006 to May 2007), there is no difference.
You have to declare the income you make in Japan the same way you do in the US. There are just different forms to file to declare your foreign income. If you are making less than $80,000/year and don't want to file for housing exception, you can use Form 2555EZ (Instructions). Otherwise, use Form 2555 (Instructions). Of course, these are in addition to your 1040.
Tax Guide for US Citizens Abroad
Other information on foreign income
I found this information from Japan-Guide particularly helpful:
"Income tax in Japan is based on a self-assessment system (a person determines the tax amount himself or herself by filing a tax return) in combination with a withholding tax system (taxes are subtracted from salaries and wages and submitted by the employer).
Thanks to the withholding tax system, most employees in Japan do not need to file a tax return. In fact, employees only need to file a tax return if at least one of the following conditions is true:
if they leave Japan before the end of the tax year
if their employer does not withhold taxes (e.g. employer outside Japan)
if they have more than one employer
if their annual income is more than 20,000,000 Yen
if they have side income of more than 200,000 Yen
Employees, who do not need to file a tax return, will have the taxes withhold from their salaries by their employer, and an eventual adjustment is made with the year's final payment.
People, who are required to file a tax return, such as self-employed persons, must do so at the local tax office (zeimusho) between February 16 and March 15 of the following year. For example, the tax return for 2006 has to be filed between February 16 and March 15, 2007."
Income Tax Guide for Foreigners