I get asked this quite alot and receive varying responses but the safe size to for designing websites in 2012 is 1024 x 768 resolution. So ideally a website should be designed to fit within this size. A good size to work with is actually 955 x 768 when designing layouts as this takes into account scrollbars and other clutter. Also, the less scrolling for important items or messages, the better. That way your audience can see anything important without having to scroll. If you need to include alot of content on your homepage then a good size for your height would be 1500/2000px.
Of course, really it is dependent of your target audience. Some people would argue that people now view the internet on larger resolutions and there is alot of truth in this if you look at the stats provided by W3C which can be found here – Browser Displays. They say that from January 2012 only 13% of users are now using 1024 x 768 and 85% view the internet on larger resolutions. 1% on 800 x 600 and 0% on 640 x 480.
On some monitors such as widescreen, then 1024 x 768 might start to look small in the center of the screen. It is possible to design a website with this in mind and so it fits 100% of the screen width, but then problems occur with layout looking lost on bigger screens and a website that might look good on 1024 x 768 starts to look less appealing to the eye.
With all this in mind, there is no absolute good size for a website, but if you follow the rule that the people are still using 1024 x 768 screen resolution then you will be listening to the current wider audience and not disregarding those that are still using 1024 x 768.
If your designing a website/intranet for a business that has all its monitors set to 2560 x 1600 then the issue is only with those computers and users and not the rest of the world. So as long as those computers are always set up to view at that resolution, then layout will be fine. If the resolution goes any smaller, scrolling will appear on the bottom of a browser.
To sum up, it is absolutely impossible to make everybody happy with website size, especially as now we have people viewing the internet on huge TV screens in their homes. There will always be somebody moaning about a website not fitting to the full width of their viewable screen or someone who can’t view a website without scrolling up and down or left to right. But you cannot please everybody.
But because people still view the internet on 1024 x 768, it is best to work with a size that fits for all and for layout and design of all modern websites. 955 x 768 (or however deep you need to go) is the best size to stick to as a rule and until people fully adapt to the larger resolutions available to them.