What is the proper designation for someone who has not finished their master's degree? If a person has completed their course work but has not submitted their dissertation or capstone project. what can you call them if they are teaching a class or course at an academic institution?
In the US, a Master's Degree does not grant a title. So, their title would be the same as if they had completed their degree. This would probably be Ms. Smith or Mr. Smith. Some institutions might have other titles. For example, a military academy would use ranks (e.g., Captain or Major Smith).

Also, the title of the person would vary by institution. My guess is the most common title would be Instructor. 

Based upon a comment by cag51, the title of Professor would also not be appropriate unless the instructor held a position with that title, which is unlikely.