99% of websites will work exactly as you expect with your Mac, with the exception of a
very few that are designed only to work with Microsoft Internet Explorer. This is because Internet Explorer does not correctly adhere to some of the agreed standards that are meant to make all sites and browsers work together. Some say this is because of Microsoft incompetence, others that it is a deliberate attempt by Microsoft to abuse their monopoly. But either way, sometimes a website made just for Internet Explorer will have trouble on other browsers.
Thankfully, this situation is now extremely rare, the last time I came across a site like this was a year or two ago and even then it was a surprise. To be honest, probably the biggest reason it is now so rare is the success of Firefox (on both PC and Mac) which means only the laziest, most incompetent of web designers would now only test their site on Internet Explorer. Also, with the new version of Internet Explorer in Vista, Microsoft have caved into pressure and tried to make it a bit more standards-compliant.
The only other situation you may come across is copy-protected video downloads (e.g. on sites where you pay to download movies). This is again because of... you guessed it... proprietary Microsoft technology. But you should not experience this in general web use where the videos are not copy-protected.
In short, there is an occasional problem but the average web user is unlikely to ever experience it. Should it be a real problem, you can always get a cheap copy of Windows and run
Parallels - the new version will actually let you have Internet Explorer load up straight from your Dock, without having to leave the Mac environment and "see" windows.
Hope that helps!