Fortunately, there are a number of variables here. First, you must accidentally happen upon a vulnerable site, then your computer must have one of several browser vulnerabilities present for the attack to take root. According to Trend Micro, the component that serves up the browser vulnerabilities is browser aware, able to infect your specific browser of choice. Assuming it can, the attack then downloads various Trojans designed to steal personal information.
To prevent such an attack, Trend Micro urges everyone to be aware of sites requiring software installation; do not allow software installation unless you trust the site and the provider of the software. Keep your PC software fully patched and be sure your antivirus protection is updating properly. And, of course, be wary of any unexpected e-mail and e-mail attachments.
For more on this specific attack, antivirus vendor Panda has prepared a 28-page PDF that provides granular detail.