Let's start with the general definition: client metrics give you an insight into the use of the system by an end user. All metrics at this level typically depend on how responsive the UI is, and how many client-side resources the application consumes.
Many modern user interfaces are browser or device based. In these situations, the primary measured values are those that are related to page views, page load time, JavaScript code, the browser or device types' geographical location, and session traces.