Australia requires Americans to apply in advance to visit the country, even if only for short periods as a tourist. Regulations, however, are quite simple and easy to navigate.
While the U.S. has added Australia to the Visa Waiver Program, allowing Australians to enter the U.S. with just their passports for up to 90 days, this doesn't work the other way around. U.S. citizens need to have not only their passports but also their visas.
The simplest way to obtain a visa is to contact the Australian government's Electronic Travel Authority, or ETA. This is done entirely online. ETA visas appear electronically on your arrival record, and nothing needs to be affixed to your passport.
The other option available to Americans is applying for an actual tourist visa--Subclass 676--at an embassy or consulate. Unlike the ETA permit, this visa has an actual fee, not just a small processing charge.
The main difference between the two options described above is their term--ETA visas are good for multiple entries, but only for three months. Full tourist visas are good for up to 12 months and can also permit multiple entries.
ETA visas cost AUS$20 to process, while Subclass 676 tourist visas have a full application fee that varies by season. For most visits, ETA application is advisable.
When planning a visit to Australia, you can apply for an ETA permit online in advance--try to apply at least a month before your travel date. You may also obtain this through your travel agent or airline. Processing times are fast; often, the permit is ready on the spot.