Ok, so this is less an article itself and more a stepping stone to other research that has been done on the subject by people far more dedicated to the world of cult movies that I.
Books
Defining Cult Movies: taking a look at cult movies from all over, Defining Cult Movies is a good starting point for trying to figure out what the phenomenon is all about.
Cult Film Stardom: Taking a look at a more specific aspect of cult movies (the cult movie stars) Cult Film Stardom outlines just what it is that turns a movie star into a cult movie star.
Articles
Far less in depth than the books listed above, here are a couple articles that can help shed some light on the subject for those unable (or unwilling) to invest the time in a book.
The BBC article What makes a cult film?
The Liveabout article What makes a movie a “cult” movie?
Videos
Unfortunately, I have no specific links here that I myself have found useful but a quick Youtube search will put you on track. Plus, places like WatchMojo and WhatCulture will have tons of lists and places to begin your search towards the truth of what makes a cult film.
Enjoy the research!