In order to be a sentence, a group of words must contain at least one full independent clause. An independent clause contains a subject and a verb and is able to stand alone.
Some sentence fragments contain a subject and a verb, but they begin with a subordinating word which makes them unable to stand alone. Examples of subordinating words include after, although, because, before, if, so that, that, though, unless, until, when, where, who, and which. These fragments can usually be easily attached to the previous sentence.
X: Darius the vampire usually preferred Type A blood. [Though
sometimes O was fine.]
C: Darius the vampire usually preferred Type A blood, though sometimes
O was fine.
X: Superman likes to go for a drink. [After he leaps tall
buildings.]
C: Superman likes to go for a drink after he leaps tall buildings.
Other sentence fragments lack a subject, a verb, or both.
X: My pet rats like to shred things. [For example, socks.]
C: My pet rats like to shred such things as socks.
X: My little brother likes to throw tantrums. [Crying and screaming
and kicking until he gets his way.]
C: My little brother likes to throw tantrums. He cries and screams and
kicks until he gets his way.
When two independent clauses are joined without a coordinating conjunction (and, but, or, nor, for, so, yet), the result is a run-on (or fused) sentence.
X: Violence seems more prevalent today than back when these books
were written this is probably not true for a number of reasons.
C: Violence seems more prevalent today than back when these books were
written. However, this is probably not true, for a number of reasons. Or,
Violence seems more prevalent today than back when these books were
written, though this is probably not true, for a number of reasons.
When two independent clauses are separated only by a comma, with no coordinating conjunction, the result is a comma splice.
X: Fanon talks about violence in Africa, Invisible Man is about
the United States.
C: Fanon talks about violence in Africa, whereas Invisible Man is
about the United States.