IE, Opera, Avant, AOL for Windows and MSN all use the IE (MSHTML) engine. This is the most popular engine and the easiest to code but for advanced coders or professional websites, so much more can go wrong. If your website works in IE, 70% of the computer world will see a perfect website. IE does not display CSS or PNGs correctly. IE 7 will fix these issues.
Mozilla, Netscape, Camino, Firefox, AOL for Mac, and all these "open source-mozilla/netscape browsers" use the Netscape Gecko engine. This is the most advanced engine for websites and little can go wrong with coding because these browsers are smart and can detect what the coding is USUALLY supposed to be. If a website is perfect in the Gecko engine, the MSHTML IE engine may not display the websites properly.
Safari and other "Apple Only" browsers utilize the KHTML engine, an engine which is sometimes critisised by users because it is the least advanced engine. It is better than the MSHTML engine in some ways but overall it is unsuccessful and a lame attempt to fight microsofts browser. However, it is always advancing and currently little behind Microsofts HTML engine. Safari 2.0 MAY have made it better than MSHTML by upgrading KHTML database, etc.
Sites that display a button for "Taking Back The Web" are usually designed for the Gecko engine. Websites that say "Made for Internet Explorer 5.5+" are, well its self-explanatory, and websites that say "Made on a Mac" are USUALLY KTHML compliant.
W3.org has a coding checker which will check the coding of your website and tell you how to fix it so it is a "universal website" for every browser. Life would be easier if everyone used Firefox/Camino.