Every sentence must have a subject and predicate (unless a subject or predicate is clearly understood or implied).
In each of the examples below, the second statement is not a sentence but a sentence fragment.
1. Old Testament stories portray two types of leader. The brainy leader and the one who relies on his physical strength.
Incorrect: the fragment is an appositive* detached from its antecedent and punctuated as a complete sentence.
Correction: join the sentence parts by a comma, a colon, or a dash after the first leader.
2. I surveyed the party. The furniture overturned, empty cans lying on the floor, and a funky smell coming from the bathroom.
Incorrect: participial phrase punctuated as a complete sentence.
Correction: change overturned, lying, and coming from to finite forms of the verbs--i.e., was overturned, were lying, was coming from.]
3. Hamlet thinks of his father's ghost frequently. Which almost makes him insane.
Incorrect: subordinate clause punctuated as a complete sentence.
Correction: join the subordinate element to the main element: Hamlet's frequent thoughts of his father's ghost almost make him insane.
See also Punctuation ("p") in this handbook.
* An appositive is a phrase appearing next to a noun that explains or identifies that noun and is equivalent to it: Dunham, a light-side dorm, is instead rather like a dungeon.