What Is a Sentence?
In its most basic definition, a complete sentence must contain a subject (noun or pronoun) and a verb. It's a "complete thought."
Sentence: Jane ran.
Sentence: It will snow tomorrow.
Fragment: Snow tomorrow.
As you can see in the above examples, the sentences have a subject—"Jane" or "It"—while the fragments don't. They're just words, hanging out, with no real purpose or context.
The subject doesn't have to be explicitly stated in the sentence; it can be understood. This type of sentence is called an imperative.
In the first, correct example, the reader understands that the speaker is commanding someone to run. In the fragment example, we have a lone verb, feeling lonely. Imperatives are more complicated in English than other languages because there's no specific verb tense to indicate it. Imperatives are conjugated as the second person present tense, without the subject. Often an exclamation point is used in place of a period, but the best clue is to ask yourself if the lone verb is urging or commanding a listener to actively do something. If so, you've got yourself a sentence; if not, you might need to rethink it.
Another type of sentence fragment is the dependent clause. It depends on the main clause of the sentence, and it can't (or usually shouldn't) exist on its own. That sounds a little scary, but here are a few examples:
Sentence: Jane ran because it was getting dark.
Fragment: Because it was getting dark.
Sentence: I'm happy that it'll snow tomorrow.
Fragment: That it'll snow tomorrow.
As you can see, the "fragment" examples are actually a part of the longer "sentence" examples. Without the main clause, the dependent clause is bereft of context. Like our lonely verbs in earlier examples, it forms part of a thought, but not a complete one.
Is it possible to play with these hard-and-fast rules of grammar? Of course it is, but tread carefully!
Using Fragments to Establish Tone
Cormac McCarthy is a master of the descriptive fragment, so let's look at a few examples from The Road:
Example 1: Across the field to the south he could see the shape of a house and a barn. Beyond the trees the curve of a road. A long drive and dead grass. Dead ivy along a stone wall and a mailbox and a fence along the road and the dead trees beyond. Cold and silent. Shrouded in carbon fog (112).
Example 2: He sat in the sand and inventoried the contents of the knapsack. The binoculars. A half pint bottle of gasoline almost full. The bottle of water. A pair of pliers. Two spoons (73).
In both of these paragraphs, only the opening sentence is complete; the rest are fragments.
McCarthy makes using fragments look easy, but they're actually tricky, and it's important to establish context before you go throwing them around all willy-nilly.
Random Lonely Fragment: Long, soft beams of moonlight.
Fragment(s) Used Poetically in Context: Morpheus, Lord of the Dreaming, perused his domain with a satisfied gleam in his starry eyes. Long, soft beams of moonlight. Eve's cave, mysterious and deep. A crooked house. A pumpkin-headed man.
In this passage, as in the McCarthy examples, fragments are being used to establish the tone or mood of the story. Morpheus is thinking in snapshots, not in complete sentences; McCarthy is using fragments to express the desolation of the landscape and the bleakness of the characters' circumstances.
Using Fragments in Dialogue
Using fragments in dialogue can establish a character's mood or it can simply be a unique way that character has of speaking. In Watchmen by Alan Moore, Rorschach rarely speaks or writes in complete sentences.
Example: Stood in firelight, sweltering. Bloodstain on chest like map of violent new continent. Felt cleansed. Felt dark planet turn under my feet and knew what cats know that makes them scream like babies in night. Looked at sky through smoke heavy with human fat and God was not there ("The Abyss Gazes Also" 26).
Moore uses the fragments to show Rorschach's unique mindset and point of view. Even if you weren't reading a graphic novel that uses specific Rorschach-dedicated typeface, you would instantly recognize Rorschach as the speaker just from the style of the dialogue.
The same applies to using fragments to establish a character's mood.
Example: "Yesterday I had to clean out the ravens' cages," Merv said. "What a job. The poop. The feathers. And why? Because the Boss wanted me to, that's why!"
Using fragments in Merv's dialogue lets the reader "hear" his frustrated tone. It's a way of indicating your speaker's mood while avoiding pesky dialogue tags and extraneous adverbs. I could have added "Merv said testily" or "Merv complained," but the use of fragments shows all of that. Stating it isn't necessary.
It's incredibly important to be discriminating with your use of fragments. Not every story will call for them. Overuse of fragments can make your reader think you simply don't know how to form complete sentences! Make sure your grasp of sentence structure is sound before you experiment with fragment use. Start with the basics, and build from there. When used properly, fragments can be a great descriptive tool. Find an author you like who uses fragments (like, hey, Cormac McCarthy!), and be inspired.
For some more information, check out the following:
Grammar 101: Basic Sentence Structure