This is something you will most likely come across if you make your own drupal theme or start a theme from scratch. Cross browser compatibility is something every designer / developer (unfortunately) have to consider.
This is mostly because of the way Internet Explorer handles CSS stylesheets. Given that IE has a 67.51% market share compared to Mozilla Firefox (21.73%), Safari (8.00%), Google Chrome (1.15%), Opera (0.71%) and 'Other' (0.90%) (Source: Net Applications) this is a given. Your page might render beautifully on all the browsers above except from Internet Explorer, which is a pain - but is unfortunately we all have to live with. So how do you set up your drupal site to tackle this? First of all you will need to get your hands dirty and edit your page.tpl.php file.
Insert this code just below your title tag:
@import "<?php print base_path() . path_to_theme() ?>/fix-ie-6.css";
@import "<?php print base_path() . path_to_theme() ?>/fix-ie-7.css";
And then make two files in your theme folder called fix-ie-6.css and fix-ie-7.css and start tweaking away on the styles that are different in those two browsers. Of course if you see the need to add another stylesheet for IE 8 do the same again for that browser.
Another tool I find indispensable in this process is http://browsershots.org/ (but of course the best thing is to have a PC handy).
Good luck!