Sentence frangments: sentence fragment: phrase or clause written as a sentence but lacking an element, as a subject or verb, that would enable it to function as an independent sentence in normative written English.

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.

A complete sentence must include:
1. a subject (the actor in the sentence)
2. a predicate (the verb or action), and
3. a complete thought (it can stand alone and make sense—it’s independent).
Some sentences can be very short, with only two or three words expressing a complete thought, like this:
John waited.

Some examples of sentence fragments:
Because his car was in the shop(What did he do?)
After the rain stops (What then?)
When you finally take the test (What will happen?)
Since you asked (Will you get the answer?)
If you want to go with me (What should you do?)
How to identify the type of fragment that you have found.
You can correct a fragment two ways: 1) adding the necessary main clause or 2) connecting the fragment to a main clause already in the passage. Whether you add or connect, you must use the right punctuation.
Some fragments, for example, will require a comma if you connect them at the beginning of a main clause. If you choose to connect them at the end, however, these same fragments require no punctuation at all. Other fragments will require a comma whether you connect them at beginning or the end. To make an intelligent comma decision, you first have to identify the type of fragment that you have.
A fragment will often be a subordinate clause, participle phrase, infinitive phrase, afterthought, lonely verb phrase, or appositive. Each type of fragment has a marker that identifies it.
Subordinate Clause Fragments
A subordinate clause fragment [sometimes called a dependent clause fragment] will begin with a subordinate conjunction, a relative pronoun, or a relative adverb. It will also contain a subject and a verb. Unfortunately, this combination of words will not express a complete thought by itself.
Think of the problem like this: At work, there are bosses and their employees, also known as subordinates. When the bosses aren't directly supervising, many subordinates goof off. In a sentence, the main clause is the boss. If