ALL visitors to Australia who do not have an Australian (or New Zealand) passport require a visa to come to Australia.  Fortunately this is cheap (and painless) to acquire.  It is valid for 12 months and allows a maximum of 3 months for any one visit.

It is linked to your Passport Number so if you subsequently get a new passport you will need to notify the Hobart Global Processing Centre with your new passport number (Postal Address: GPO Box 794, Hobart TAS 7001, AUSTRALIA).

Go to http://www.eta.immi.gov.au/

Click on "Am I eligible?" to double-check that you can acquire a visa electronically.  Those from EU countries and the USA are definitely eligible.

The cost is AUD20 (USD16, GBP8) and can ONLY be paid via credit card (American Express, Diners Club, JCB, Mastercard and Visa).

Click "Apply for ETA" (Electronic Travel Authority) and follow the instructions (you'll need your passport).

If you are NOT on the eligible list you will need to fill out a Form 48 (or 48R) to gain a Tourist Visa TR 676 and visit an Australian consulate or High Commission in person.