What is the standard for screen sizes when designing web sites? Is it still 800x600?
The standard is to make your CSS flexible (fluid) and not rigid (or kept to a specific width), so that it doesn't matter what size desktop your site is being viewed on, it can be seen in its totality, without annoying space gaps.
960 pixels
seems to be the current rage for width, though.
With the laptop and LCD monitor explosion, its safe to go with 1024 and be prepared to be seen in 1280-1440+ by a good portion of the web. The fixed width trend atm is the 960 pixel grid. But I'm a bigger fan of fluid layouts.
1024x768 is the lowend res generally targeted these days. usually with a live area of 950-960 wide.
I also go with 960 for width and about 600 for height. Google Browser Size is a pretty neat way to visualize what you can expect your visitors to see.
There is no one standard and there never has been. You have to bear your audience in mind. Do you have people accessing on a PDA with a very narrow but possibly zoomed width? Do you have people accessing on a netbook with a generally quite small resolution? Are they newbs wasting half their screen width on a sidebar? Do you have people with 30" high-res displays browsing full-screen?
So then you make a flexible layout with liquid parts that respond to browser width, and look acceptable across a range of resolutions which includes all those you said ‘yes’ to above. (And hopefully they'll still remain usable for resolutions outside those ranges, if not exactly pretty.)
For what it's worth, most of my mainstream (ie., not mobile-focused) sites remain more-or-less as-designed from widths from around 700 to around 1900. But your users may differ.