There are not too many sites that won't work with Firefox and Opera these days. But for the odd ones that won't there is that extension that loveoflearning has highlighted.
The reason some sites won't work is all to do with HTML standards. Firefox, Opera and other browsers interpret the code that lies behind every web page in a manner laid down by the
W3C. Microsoft has added capabilities to their browser that allows it to interpret non-standard code. Web site writers use this to add web page effects that only work in IE and sometimes use code that other browsers don't understand. Because Internet Explorer is the most dominant web browser in the world, some web page writers are not concerned if their pages don't look right in other browsers.
There's two sides to the argument. Some say that the alternative browsers should cater for non-standard code and others argue that site writers should stick to W3C standards to ensure pages work properly. In the middle are the Firefox (or other) browser users who don't care about all that and just want
all pages to display properly.
If you come across a site that doesn't look right in the browser of your choice, you could get in touch with the webmaster and point this out. Some ignore the e-mail but some reply saying their site is due for an update and that part of this will include modifications to ensure compatibility with all browsers.
I think there is a general awareness now that millions of people are using browsers other than IE. Webmasters perhaps can't (or shouldn't) ignore that now. Many Firefox users don't bother finding ways of looking at sites that don't display properly. They just go onto somewhere that does. For a business site this must represent a fair number of lost sales. For those who just want people to visit or use their site, it could result in a loss of visitors.