No, there isn't one, but usually I made mine around 900 px wide, considering that there are still a lot of people using a 1024x768 display. The height depends on the content and the design of the page.
There definitely are not "standards" per se, but targeting your users by understanding their screen resolutions is the best place to start as MJ080 points out; 1024px width would be a good starting point.
That said, your design needs to consider variability in how the browser is fit within the monitor, and how your site would fit into the browser page itself. For this reason, you need to think about browser chrome (like vertical scroll bars), etc. This is why you tend to see a lot of sites built under 970px (or so) page width. There are also an abundance of CSS frameworks out now that you can use as building blocks, such as http://960.gs.
Vertical space is not as limiting because users are accustomed to vertically scrolling for additional content. That being said however, not all page real-estate is created equal. You'll obviously want to keep your most important content toward the top, and above the "fold" (where the page content gets cut off due to browser height/screen resolution).
Lots of documentation online, just search around and read up on best practices and such.
There is still about 3% out there that are still using a resolution of 800x600. Taking into consideration the amount of screens world wide then 3% is a pretty large number.
Users will scroll vertically all day but refuse to scroll horizontally and will quit your site rather than do so. If you're trying to make money from your site then do you really want to risk losing those potential customers?
So really you should be aiming for about 770px minimum width to make sure all visitors can view your site properly. This is why I always use a liquid design.
No there is no standard size for a website. It depends on your site purpose, content, design and layout. You need to be aware of the "graphic safe" areas while you design.
Though most display screens have large resolutions, there are many many people who still use 800 X 600 resolutions. Web page graphics that exceed the width dimension of the most common display screens look amateurish and will inconvenience many readers by forcing them to scroll both horizontally and vertically to see the full page layout. It's bad enough to have to scroll in one (vertical) direction; having to scroll in two directions is intolerable.
Graphic "safe area" dimensions for layouts designed for 800 x 600 screens:
Maximum width = 760 pixels
Maximum height = 410 pixels (visible without scrolling)
Or go for the liquid design which should generally keep most people happy. But there are limits to what you can achieve with it.