Back in the day, 800 x 600 was the screen resolution to design for - and maybe 640 x 480. Then along came 1024 x 768, etc, etc, etc.
But then it gets worse: now we have
Watch out for high DPI settings
I think one of the most undiscussed issues facing front-end web development today is testing on high DPI systems. Everyone has learned to test and test and test on different browsers but designers/developers have not caught on to testing on different DPI settings.
High (or even low, for that matter) DPI settings break designs when fonts are scaled but images are not (which can happen), can cause images to look fuzzy/blurry, and absolutely positioned objects may not appear in the desired location (which would be devastating for CSS menus.) If nothing else, test your images at high DPI and re-render them as necessary.
This has never really been an issue until recently with the release of Windows 7 and people buying computers with high resolution monitors. First off, Windows 7 uses 96DPI as default (which is different from the rest of the computing world that has been using 72DPI as the standard. Moreover, Windows 7 will automatically adjust DPI settings and I have seen people with DPI of 150% of normal (96 DPI in Windows).
Here's a great link discussing this issue in more details: http://webkit.org/blog/55/high-dpi-web-sites/
A great cross-browser website designed with web standards is the goal but don't forget about DPI testing.