I worked for the United Nations in the United States and am an American citizen. Do I pay both self-employment tax and income tax, or just the former? Do I pay state tax?

Your United Nations income is subject to Federal income tax and self-employment tax. It is exempt from state and local taxes.

However, the United Nations will reimburse your Federal tax and half of your self-employment tax

http://www.un.org/Depts/oppba/accounts/tax/faqs/faq03.htm

You only need to pay SE tax when you work for the UN in the USA. You are not required to pay SE tax when working for the UN abroad, but, on the same token, you won't be contributing toward your social security.

You are required to pay state and local tax as well as federal tax (and social security tax while stateside).  If you move abroad you may not have to pay state and local taxes if you meet certain criteria. Also note that your social security benefits may be subject to the windfall elimination provision if you work abroad and don't have 30 years of substantial earnings and of paying into social security.

This answer isn't wholly correct.  US citizens employed by the UN are NOT exempt from State and Local taxes. One must file with NY State and also the state in which one lives (if not in NY). If one lives in New York City or Yonkers, those also must be accounted for. The UN provides reimbursement for State and local taxes, but one is NOT exempt.

Re: I worked for the United Nations in the United States and am an American citizen. Do I pay both self-employment tax and income tax, or just the former? Do I pay state tax?

What is  the support basis for not paying SE tax when a US citizen and resident works for the UN abroad?

How does one avoid paying this tax, since it appears that any 1099Misc entry defaults to paying SE tax?