Thanks for asking this great question! Seriously. It is an interesting question, because in the English language rules are broken all the time!
Google this: poetic license
When you write a paper for school, it is best to always use complete sentences. If you use poetic license and write a fragment, you might make the reader think you do not know the rules of grammar. However, if you blog on the Internet or write a story, you really can use fragments when you want to as long as it sounds good.
This is not good: I like to eat pie. My favorite food.
This is complete: I like to eat pie, my favorite food.
This is okay sometimes: I like to eat pie, because it is the greatest food. Blueberry Pie.
But really, you should not use fragments in formal, academic writing.
