A fragment is essentially a piece of a sentence. A complete sentence needs a subject, something that performs an action in the sentence, and a verb, the action that is performed.
Here are some sample fragments for you:
A piece of cake
This is a fragment because there is no verb. We do not know what this piece of cake is doing.
Ran to the corner.
In this fragment, there is no one running to the corner.
Having a very bad headache.
Here, we have a verb form and a noun, but this is a fragment because we have no idea who has a bad headache or what the person with the headache is doing about it.
One point I want to make here is that it is not simply a question of having a noun or pronoun and a verb. If the noun or pronoun is not doing anything, or if the verb is not giving the noun or pronoun some action, you will still have a fragment.