Activity completion allows the teacher to set completion criteria in a specific activity's settings. A check (tick) appears against the activity when the student meets this criterion. The criterion might be viewing, receiving a certain score or a student marking it as complete.
Here is a PDF summary table of activity completion conditions, courtesy of Dag Klimas.
A teacher may mark an activity complete on behalf of a student from the Activity completion report if they have the capability Override activity completion (which is enabled by default.)