Programs are passed to the computer that the browser is on, and that computer runs them. The alternative is server side, where the program is run on the server and only the results are passed to the computer that the browser is on. Examples of this would be PHP, Perl, ASP, JSP etc.
Hence the feared browser incompatibilities.
It’s best to solve compatibility problems on a case–by–case basis. In fact, most pages on this site have been written precisely because of browser incompatibilities. So read on to understand more. But I warn you: you need to digest quite a lot of information. Therefore it’s best to solve the problem at hand and leave the rest of the information alone until you need it.