The requirements for each country are different, including whether you can apply for your visa via mail, via a visa agency or whether you are required to appear in-person at the Consulate or Embassy to submit your visa application or pick up your passport with the visa once it is processed. Even once you have gathered all of your documentation and completed the visa application, consulates generally require 4-6 weeks for processing. Some consulates require up to 3 months for processing.
You should read carefully to determine which type of visa you need. Visa types include: tourist, business, and student. Some European countries will allow U.S. citizens to visit for short periods of time without a visa such as study abroad programs during the summer for less than 90 days. First, you should determine whether or not you will need a visa to enter your country of choice before you apply, since not all countries require a visa.