A complete English sentence must have both a subject and a verb to express an idea. A sentence fragment lacks either a subject or a verb.
◊ I went downtown. To buy a pair of shoes.
In the above example, the second item is a sentence fragment because it lacks a subject. Who is buying the shoes in the second item? The subject is not mentioned.
Often the best solution to sentence fragments is to combine them with sentences near them. In the above case, I went downtown to buy a pair of shoes would be a perfectly acceptable sentence (provided the writer ended the sentence with a period).
◊ We approached the summit at dusk. Dead exhausted.
A complete sentence constructed from the above might read: Dead exhausted, we approached the summit at dusk. Notice how “Dead exhausted” modifies, or describes, the subject of the clause that follows.