A Fragment - is 'an incomplete sentence'. In other words, it's a sentence without a subject or a verb. Find out more about what makes a sentence complete, and what a subject is, here.
What is the effect of using a fragment?
 Sometimes this gives the effect of confusion, ragged thoughts. The incompleteness of the utterance or phrase can create mystery, which increases suspense.
e.g. My leg!
Here, we know something's very wrong with his leg, but we don't know what.
 In other places, fragments are used for emphasis.
e.g. Or so they thought.
Can I use fragments in my own writing?
Yes. It's useful for character thoughts, direct speech, short stories, and even writing to argue and persuade (in moderation). Just like exclamation marks and ellipsis, though, less is more.