The short answer is liquid design. I've also heard it referred to as fluid/flexible/no resolution design.
In general, you would not use fixed width's in your tables, or divs. There would be exceptions, such as columns used for navigation or gutters, that would have a fixed width.
As a side note, since you're now testing your websites at different resolutions, be sure to check them out in different browsers. That could be another shock.
Test your web design in different browsers - Browsershots will show your pics of your website using multiple browsers and operating systems, and you can set the image resolution from 640 pixels wide to 1600 pixels wide.