Sentence Fragment

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. There are several reasons why a group of words may seem to act like a sentence but not have the wherewithal to make it as a complete thought.

It may locate something in time and place with a prepositional phrase or a series of such phrases, but it's still lacking a proper subject-verb relationship within an independent clause:

In Japan, during the last war and just before the armistice.

This sentence accomplishes a great deal in terms of placing the reader in time and place, but there is no subject, no verb.

It describes something, but there is no subject-verb relationship:

Working far into the night in an effort to salvage her little boat.

This is a verbal phrase that wants to modify something, the real subject of the sentence (about to come up), probably the she who was working so hard.

It may have most of the makings of a sentence but still be missing an important part of a verb string:

Some of the students working in Professor Espinoza's laboratory last semester.

Remember that an -ing verb form without an auxiliary form to accompany it can never be a verb.

It may even have a subject-verb relationship, but it has been subordinated to another idea by a dependent word and so cannot stand by itself:

Even though he had the better arguments and was by far the more powerful speaker.

This sentence fragment has a subject, he, and two verbs, had and was, but it cannot stand by itself because of the dependent word (subordinating conjunction) even though. We need an independent clause to follow up this dependent clause: . . . the more powerful speaker, he lost the case because he didn't understand the jury.

Stylistic Fragments

There are occasions when a sentence fragment can be stylistically effective, exactly what you want and no more.

Harrison Ford has said he would be more than willing to take on another Indiana Jones project. In a New York minute.