A sentence must include both a subject and a predicate. The subject is based on a noun or pronoun; the predicate is based on a verb that describes what the noun or pronoun does or has done to it:

Helen [subject] opened [predicate] the book.

A sentence may evolve a great many complexities, but the basic structure will still be visible:

Helen [subject], a victim of morbid curiosity, tore open [predicate] the book and recoiled in horror from what she saw.

A sentence fragment lacks a subject, a predicate, or both:

Went on to become a political leader. (no subject)

Plato, the eminent Greek philosopher. (no predicate)

While traveling to Sparta.

The last example is not promising as either subject or predicate; "traveling" is a verb form, but it can't function as a verb because it cannot, by itself, express the action performed by a subject: one would say, "He travels to Sparta," but not, "He traveling to Sparta."

A sentence fragment cannot stand alone and still express a complete and fully comprehensible thought. This is obvious in each of the three examples given above, but it is also true in the following case:

Because he had not been trained for the job.

If this statement were limited to "He had not been trained for the job," it would be a complete and correct sentence; it has a subject ("He") and a verb ("had been trained"). The "because," however, points to some other statement on which all of this depends; "because he had not been trained for the job" is in fact a dependent clause (see Handbook, pp. 26-28, 34-35) that can make sense only when something else is added:

He [subject] failed [predicate] because he had not been trained for the job.

There are only a few situations in which sentence fragments may properly be used. They are sometimes effective in providing special emphasis--

There was no one there. No one at all.

--or in giving brief answers to rhetorical questions:

Is this an intelligent decision? Hardly.

(For other instances in which fragments may be used, see Handbook, pp. 34-39.)