Like the brainstorming that goes into the development of a site's purpose and goals, user testing can be an important part of the development process. This is particulary true when developing on a broad scale (as opposed to defining the particularities of a site and user behavior.) Differences between intentions and actual behaviors can sometimes be found.
Depending on the time and scope of the testing schedule, here are two approaches:
Selecting who to test can be problematic. Your target audience is the ideal subject. Any mailing (electronic or otherwise) database that you have available would be a prime starting place. Your assessment of who your audience is might point to a number of related organizations with likely candidates. There are newsgroups and mailing lists that may have interested parties. But outsiders, people not planned for, can be useful too. Quite often, though, some sort of compensation or gift is usually required in order to get anyone interested enough in your efforts to take the time out of their day for you.
Task analysis, as this area is sometimes called, should be done, ideally, by someone other than those who created the site. While it can be done this way, it is generally better especially for large-scale projects to have a professional consultancy handle this. In either case, getting a non-biased and scientific result requires rigorous methodology. Beyond focus groups and individual testing (utilizing screen captures and various tape recording devices), things like usage tracking and areas for questions and suggestions can be incorporated into the actual site.
Once you have the results of your surveys and suggestions, look for patterns of reactions. If everyone is pointing to some awkward construction, its time to go in there and fix it! And while not every response and issue can be resolved, it has been suggested that an 80/20 rule be applied towards the constant construction that is the nature of web design.
Here are some user testing resources: