Americans do not need a visa if they plan to visit the Czech Republic for less than 90 days. However, any foreigner that enters the country needs to have a passport that will be valid for at least the next three months, proof that they can cover the expenses of the trip (a credit card or bank statement usually works) and proof of health insurance that can cover any complications while in the Czech Republic.
Aside from the U.S., citizens of these countries do not need a visa to get into the Czech Republic (but everyone needs to meet the tourist entry requirements): Andorra, Antigua and Barbuda, Argentina, Australia, Bahamas, Barbados, Brazil, Brunei, Canada, Costa Rica, Guatemala, Honduras, Chile, Croatia, Israel, Japan, Macedonia, Malaysia, Mauritius, Mexico, Monaco, Montenegro, New Zealand, Nicaragua, Panama, Paraguay, Salvador, San Marino, Serbia, Seychelles, Singapore, South Korea, Uruguay, Vatican and Venezuela.
If you are a citizen of a country not on the exemption list, you should apply for a "short-stay visa" at a diplomatic mission in your country several months before your trip. For this you need an application form (available on the Ministry of Foreign Affairs of the Czech Republic's website), passport, photograph, document establishing purpose of stay (letter of invitation, itinerary, etc), hotel reservation, proof of ability to cover expenses and proof that you will leave (return plane ticket, proof of family and work ties elsewhere).
Students planning to study for more than 90 days in the Czech Republic need to apply for a student visa. You need to submit your passport, an application form, a photo, a document confirming that you are studying in the Czech Republic (a document confirming enrollment or intent of study from a school or the Ministry of Education), a document confirming accommodation, a document from the police stating any criminal record or lack thereof (translated into Czech), a medical report, medical insurance that can cover you in the Czech Republic and proof of payment of the visa fee. Apply early, this can take up to three months.
A citizen of any country who wants to stay and work in the Czech Republic needs to apply for a work visa. You need to submit your passport, an application form, a photo, a document confirming that you have work in the Czech Republic (employment permit from the labor office or proof that the permit is under review or not necessary), a document confirming accommodation, a document from the police stating any criminal record or lack thereof (translated into Czech), a medical report, medical insurance that can cover you in the Czech Republic and proof of payment of the visa fee. The State Department recommends that people apply several months in advance of their trip.
