A sentence fragment is an incomplete sentence. Some contexts require sentence fragments (annotations in annotated bibliographies, where the implied but omitted subject is the article/book you are annotating; university course calendars, where the implied but omitted subject is the course being described). In most forms of academic writing, however, we need to write in full sentences that have a subject, a verb, and (in most cases) an object.
Wrong: Defined in the OED as “The garden plant called honesty.” [This fragment lacks a subject.]
Right: “Lunaria” is defined in the OED as “The garden plant called honesty.”
Full sentence turned into a fragment by addition of a conjunction
Right: Adam ate the apple.
Wrong: Because Adam ate the apple.
Fragment created by punctuation error
Substituting a semicolon for a comma may also create a fragment.
Wrong: Adam worked in the garden; content with his role.
Right: Adam worked in the garden, content with his role.