A sentence fragment is a group of words which sounds like a sentence but does not express a complete thought. Sometimes it has a subject or verb missing. Sometimes it is a phrase or subordinate clause instead of a complete sentence.
Sentence fragments are normally not only grammatically incorrect but also difficult to understand.
The one type of fragment which may be acceptable at least in conversation is a fragmentary response. This is a sentence fragment which answers a question or otherwise responds to a situation in such a way that its meaning is understood even though it is not a complete sentence.
Fragmentary responses should be avoided except in dialogue or very casual writing.
Example:(The response is not a complete sentence, but it is understood because it is responding to a specific question.)"Who was that woman with you last night?"