Unfortunately the major browser developers (Microsoft, Mozilla, Opera, Apple, etc) have been kinda' reluctant to come to a real agreement on web standards. Still they have come very far in this matter compared to what it used to be a few years ago.
Today their respective browsers (their latest versions) are almost 100% percent cross-compatible. The World Wide Web Consortium (or W3C) has done a very good job mediating among these giants and has established the XHML standard, hopping that each of these companies put their individual interests aside and adopt a better accepted and compatible way of rendering web pages.
So today, as long as you develop complying with the latest W3C standards, you have a better chance for your sites to be cross-browser compatible.
I suggest you starting with the XHTML vs HTML tutorial at:
http://www.scriptingok.com/tutorial/XHTML-vs-HTMLREGARDS!!