The course completion status block shows what has been done towards completing the course. The student and teacher will both see this block. The "More information" link will lead to a more detailed report. This report can have some interactive elements in the form of check boxes for the student and teacher.
This block shows the status for each student. It is also the place where the teacher (or others) can mark the course as complete.
Note: The course completion status block will only appear if completion tracking criteria are set in Settings > Course administration > Completion tracking. This is what the teacher sees: