If you have an official document that you want to use in another country, you will probably need to have it certified. Documents that typically need to be certified include vital records (e.g., birth, death and marriage certificates), court judgments and corporate records.
Countries that have signed the Hague Convention Abolishing the Requirement of Legalization for Foreign Public Documents require a relatively simply process known as apostille (similar to having a document notarized). Countries that have not signed the Hague Convention require a more complicated process known as "authentication" or "legalization."
The Hague Conference on Private International Law posts the text of the Hague Convention Abolishing the Requirement of Legalization for Foreign Public Documents, a list of the participating countries, a list with the certifying agencies for each country, and information on how to get an Apostille.
You can find a country-by-country and (for the U.S.) state-by-state summary of the applicable rules in A Practical Guide to Document Authentication: Legalization of Notarized & Certified Documents (Oxford University Press).
In the U.S., many official documents are certified by the State Department's Office of Authentications. States and courts generally certify their own documents. If you are in another country, you can go to a U.S. embassy or consulate. For more information, see the Notarial and Authentication Services page posted by the Bureau of Consular Affairs.
If you need to have a foreign document certified for use in the U.S., that can usually be handled by the foreign country's embassy or consulate here.
Many companies specialize in in getting documents authenticated/legalized or "certified by Apostille." NCR-National Corporate Research (800-494-5225) does this work at the State Department and embassies in the D.C. area.