Instruction

Click on Edit next to the name of your activity, and Choose the Edit Settings line. You will find an Activity Completion section in the middle of the page that opens.

Remember that there are three options in Activity Completion:
Do not indicate activity completion - no checkbox close to the activity
Students can manually mark the activity as completed - students tick the checkbox themselves to change it. (Note: they can do this even without doing the activity!)
Show activity as complete when conditions are met - the selected completion criteria must be met before the tick appears in the checkbox.

The third case needs the teacher to choose completion criteria, which can be quite different depending on the activity or resource:
1. Require view: this one is the simplest one. The student must open the resource to get a completion tick.
2. Require grade or passing grade in a quiz: in the first case, they just need a grade, whichever one they get, while in the second case, they must have a grade higher than a fixed threshold. In this latter case, the quiz completion on teacher side will show either green ticks (quiz completed and passed), or red crosses (completed but failed).
3. Require full reading or time spent in Lesson settings: these settings are specific to the Lesson activity. The teacher can choose "Student must reach the end of Lesson page" and/or add a minimal time spent on the activity in minutes. If they finish the lesson sooner, they will see an alert saying they did not reach the minimum time acceptable and may have to repeat the lesson.
4. Require submission or grading for an assignment: an assignment can be considered complete when  students submit their assignment, or only when they get a grade from the teacher.
5. Require posting, creating a discussion or replying in a forum: in the first case, a student must post, either by starting a discussion or by replying to an existing discussion. The two other options are more precise, asking for creating a discussion or replying to someone else.  In all cases, the total number of posts can be specified.

In all cases, the teacher can add an expected date to complete the activity or resource. 

You can get a more detailed information on the Moodle docs webpage on Activity Completion.

You may find interesting in this Moodle Learn video on Activity Completion.