United States citizens will automatically be granted a tourism visa at their port of entry, whether it be by land or air. The only required documents are a U.S. passport valid at least six months past the date of entry, and tickets or documents for return or onward travel; it is extremely rare that the authorities will ask for anything other then your passport. You may extend the original 90 days by an additional 90 days at the immigration office in Buenos Aires, or by leaving the country and returning. A maximum of two tourist visas, or 180 days per year, is permitted.
A business visa can be requested at the Argentina embassy or one of its consulates in the U.S. An appointment can be made by phone. The required documents are the same as for a tourist visa, plus a letter from your employer stating your business; they may also ask for financial records. All documents must be translated into Spanish and notarized.
An official or diplomatic visa can only be obtained at the embassy. You must make an appointment and bring your passport, one passport-size photo and a letter from the department or agency that sent you. The letter must contain your name and occupation, the purpose of your trip, the dates of your trip and a contact person in Argentina.
All other visas must go through the embassy or consulate; contact those entities for an appointment and more information. Most visas can be obtained within Argentina as well if you enter on a tourist visa. It may be easiest to apply for residency once you are in country if you plan to stay for an extended period of time.
Passport holders from European or South American nations have similar or identical requirements. A list of nations that do not need to apply for a tourist visa can be found at http://www.learn4good.com/travel/argentina_visa.htm.
