What is a fragment in writing?

A fragment is a piece, a small part, a shard. In A Christmas Carol, Scrooge talks about a "fragment of underdone potato".

In writing, a fragment is a piece of a complete sentence. It expresses less than a complete thought. Our speech is filled with fragments, because when we are talking we move from thought to thought without finishing them. We can understand it because when we are talking we have the opportunity of asking for the speaker to complete his thought. That is not possible with writing.