In the context of the web, a usability test is any organized use of a website to test the usability of the design. Often a usability test involves a test subject that is unrelated to the project, and who has no previous knowledge of the website topic or design. Also, it is common to video and audio record a test session, for later viewing and analysis.
The purpose of a usability test is to determine if design choices are actually user friendly and usable for a common website visitor. Often witnessing someone who is unrelated to a project use a website for the first time is very telling, and gives developers a lot of feedback about their design choices. Often developers will run a usability test with multiple test subjects in order to gather a wider range of feedback, and find more problems/bugs.