Most of us know the difference between a complete sentence and a fragment. According to a source at Empire State College in New York, complete sentences "convey complete thoughts" and sentence fragments "unless artfully used, suggest that your thinking is fragmented, because you're only presenting a piece of a complete thought. Most of the time you don't want to convey the impression that your thinking is incomplete."

This guidance is simplified and genuine, neglecting an explanation of certain prescriptive syntactical rules. Connecticut Community College, however, offers a little more information.
Quote Originally Posted by CCC
A SENTENCE FRAGMENT fails to be a sentence in the sense that it cannot stand by itself. It does not contain even one independent clause.
Now we're getting into rules like subject-verb relationship, and vocabulary, such as clauses and whether they are dependent or independent. This gets dizzying. Too many rules. And mind you, I'm not even scratching the surface.

Although formal writing tends to shun the use of fragments, they serve a number of purposes both in fiction and nonfiction. Perhaps the most important purpose a fragment serves is emphasis. Although technically a sentence fragment is always wrong, fragments are used in a wide array of literary environments.

A simple complete sentence has a subject, a verb and an object. In English it's practically impossible to complete a sentence without a preposition as well. Consider the following.

I went to the store.

I (subject) went (verb) to (preposition) the (article) store (object).

Although simple, the above example is a complete sentence. Fragments in professional writing tend to omit the subject. But typically the use of a fragment is artistically placed within a paragraph, one wherein the reader is already familiar with the literary environment, namely the context the subject provides. Consider the following example and the placement of fragments for artistic purposes. (I will underline the fragments just in case.)

He pulled on his shoes. He wanted to run, as far away as possible. To the garage. To the mall. Anywhere his parents could not find him.

In that short example I used three fragments, each connected to the previous in order to provide some emphasis. The use of fragments not only emphasize the fact the character will run anywhere, but there is specific identification of each locale to the character. If I wrote the same thing in all complete sentences, the effect experienced by the reader would change. And certain verbs could be installed to change the meaning of each sentence. Consider the following three examples. The first two I merely merge the fragments into a single sentence. The final example I use complete sentences.

He pulled on his shoes. He wanted to run, as far away as possible. He would run to the garage, or the mall, or anywhere his parents could not find him.

He pulled on his shoes. He wanted to run, as far away as possible. He wanted to run to the garage, or the mall, or anywhere his parents could not find him.

He pulled on his shoes. He wanted to run, as far away as possible. He wanted to run to the garage. He wanted to run to the mall. He wanted to run anywhere his parents could not find him.

Each of the previous examples changed the original, and each example conveyed something different. Which do you prefer? The one early above? The first example just above, or maybe the second? It doesn't rightly matter, because what we're dealing with here is style.

Using sentence fragments is perfectly acceptable stylistically. But it's wise to make sure you use them anchored in context, surrounded by complete sentences. You could certainly start a paragraph with a fragment, but you wouldn't want to make one up of nothing but.

Use your best judgment. Part of writing well (and even breaking conventions) comes down to cadence, to flow, and more. Before ever submitting your work it is wise to read it aloud. Do those fragments add to what you're conveying, or do they only stunt it?