Usability testing involves a representative set of real people (users) trying to accomplish actual tasks on your software, websites, mobile apps, or devices. Through observing both what users do and say we are able to both quantify the experience using metrics and understand the problems in the experience.
A moderated usability test involves having a trained facilitator administer tasks and record behaviors and comments users make as they attempt to complete the tasks. Moderated testing can be done in-person or remotely. In remote usability testing, participants log into screen-sharing software (e.g. GotoMeeting) and complete tasks on a computer or mobile device.
Participants are often asked to think aloud as they complete tasks, although the facilitation style changes depending on the research goals. Moderated studies can have an emphasis on finding and fixing problems (formative evaluation) or can be metrics and performance based (summative or benchmark evaluation).
The downside to moderated testing (compared to unmoderated testing) generally means higher costs, longer time for recruiting and executing, and a smaller sample size.
More on comparing usability testing methods.