You can change Activity Completion settings by going all the way down to the bottom of the settings page in any activity. The activity completion options vary based on the type of activity. For example, if you have a quiz, you have choices in the drop-down menu not to show activity completion, to allow students to mark the activity complete, or to have Moodle mark the activity complete once some condition is satisfied. As seen below, the condition may be a view, any grade received, or a passing grade (this must be set in the gradebook).

For a discussion forum, you can set the number and type of postings a student must contribute.


Activity completion options for a discussion forum.

For an assignment involving a drop box, the activity can be marked complete by Moodle as soon as the student submits, or once the activity is graded.


Activity completion options for an assignment page with drop box.

 

 You may also have assignments where students are sending you in some physical object. These may be set in such a way that submissions are not allowed in the assignment page, Moodle does not have an option for the teacher to manually mark a student's activity complete, so the best option I have found is to choose Student must receive a grade to complete this activity, and then enter the students grade once you have marked the assignment. This is inconvenient in that 

you cannot open the next activity immediately upon submission of this one. You may wish to have a short section of work that is already open, and have the completion of this assignment open up the section after that one. Remember that you can always go into the Access Restrictions on any

activity, and manually allow or disallow a specific student from accessing that content.