Conventional sentences are complete thoughts with a subject and verb (although either may be implied). But the formation of a sentence is a purposeful act of the writer. Writers will occasionally construct a fragment for a stylistic purpose. [EDIT]
Particularly in academic and scholarly writing, the conventional expectation for sentence formation is the complete sentence. In published writing of all kinds, however, many writers use occasional fragments for effect in their writing. When a writer incorporates fragments purposefully, they are completely acceptable; most fragments in student writing are assumed to be errors, not the intent of the writer. All writers, students and scholars seeking publication, must be careful that any use of a fragment is clearly a purposeful decision by the writer.
An on-line resource for sentence formation and fragments is available here: