It comes as a surprise to many Americans that we need visas to travel to some countries. At least half of the people who are surprised are simply unaware that visas even exist; usually, these are folks who either don't have passports (a huge number of Americans) or who got them only for a trip to Europe or, more recently, Canada.
But among those who do understand the concept of a visa, a relatively large number of Americans whose travels haven't taken them to visa-requiring countries assume that American travelers are exempt from visas. Their logic is that since we come from the most advanced nation on Earth, it'd be extremely unlikely that we'd be going somewhere to try and slip onto the social welfare roster or take away a job from the local populace.
It's not bad logic on the surface, because Americans tend to view control over immigration as a way to keep people off of social services. But it's wrong.
Americans need visas to visit a range of countries, including a few developed nations like Australia but also the larger developing countries that our media regularly cites as industrial powerhouses -- Brazil, Russia, China, and India -- and an array of smaller developing nations.
For the most part, it's not that these countries worry about Americans slipping into their countries unnoticed and lining up for free food or subsidized housing. Actually, it's precisely that Americans have money. So, if they want to visit, why limit the take to whatever sales taxes are collected on their purchases? It's easy to collect additional money by requiring a visa.
There's also a degree of tit-for-tat to the process. Americans pay more for visas in Brazil, for instance, than do the citizens of many other countries, because the Brazilian government decided to charge American visitors the same price of a visa that the American government charges Brazilians. Makes sense, right?
You can look up the visa requirements for any country based on the passport on which you travel using sites like VisaHQ. Americans are often better served by the Department of State's travel pages, though, because in addition to the visa requirements that Americans in particular have for each country, these pages also include travel advisories and medical information.
In most cases, when a visa is required, it means filling out paperwork that includes when and why you intend to visit, providing two of those passport-sized photos, and paying the fee. Some countries may require interviews at their consulates or other security measures. Visas of this sort are usually pasted into one's passport.
There are also countries that provide visas either electronically (as is the case with Australia) or upon arrival (such as Bahrain). These visas are included in the entry stamp (which is really unto itself a visa, demonstrating permission granted by the border officer for the traveler to enter).
In a few rare cases, visas may need to be processed outside of the United States; for instance, Americans are not forbidden by the U.S. government to visit the Democratic People's Republic of Korea, but there isn't a North Korean embassy in the U.S. Tours of North Korea are tightly organized, though, so in these instances, the process of getting visas will almost certainly be handled by the tour provider.
Whenever you travel, it's absolutely essential that you know whether you'll need a visa and what kind of visa you need. For most people, the basic tourist visa is sufficient and appropriate. Anyone planning to have a job of any kind while in a country will need a work visa. Students generally have their own special visas, and some countries regulate the presence missionaries through visas issued solely for religious work.
You should photocopy any visa that is pasted into a passport in advance of travel. Leave one copy left at home in case there are problems and another carried on you while you're in country. This is especially critical in countries that require an exit visa, without which you will not be allowed to cross the border to go home.
One piece of good news for American travelers: we usually have access to multiple-entry visas, which are good for access for an extended period of time that may range as high as five or ten years from the date of issuance.
There's often a minimal price difference between a single and a multiple-entry visa, so save yourself the hassle and get the longest-duration visa you're allowed to obtain. You never know when you might have the chance to go back!
 Travelers to the Chinese mainland require entry visas. Hong Kong and Macau are Special Administrative Regions (SARs) that at this time do not require visas for American travelers. Travel to Tibet requires special permission beyond an entry visa.