A sentence fragment lacks a subject and/or a verb, or is a subordinate clause, but is part of the sentence.

The Phrase Fragment

A group of words that lacks a subject and a verb is known as a phrase.  You will use several kinds of phrases as a writer, but a phrase must be attached to the sentence in which it belongs.  Alone, they are considered to be sentence fragments.

FRAGMENT    I am graduating in May.  With a degree and a sense of accomplishment.
SENTENCE    I am graduating in may with a degree and a sense of accomplishment.

FRAGMENT    My pet goat Stumpy only has three legs. But a surprisingly fast gait.
SENTENCE    My pet goat Stumpy only has three legs, but he has a surprisingly fast gait.

The Subordinate Clause Fragment

A subordinate clause is a group of words that contains both a subject and a predicate, but because it is introduced by a subordinating conjunction (i.e. although, until, while, that, as if, since, because), it must be attached to a sentence.  If you can identify a subordinate conjunction, you will have no problem detecting and correcting these fragments.

FRAGMENT    I had to stay up all night writing my English paper.  Because I took a nap all afternoon.
SENTENCE    I had to stay up all night writing my English paper because I took a nap all afternoon.

FRAGMENT    Bryan makes me laugh. Since he knows many good jokes.
SENTENCE    Bryan makes me laugh, since he knows many good jokes.