A fragment is an incomplete sentence; it lacks something to complete the thought. The most obvious fragments are missing either a subject or a main verb.
The following is a fragment: Hiding a rabid gorilla in Becky's dorm room.
This has a verb, but no subject. It leaves you wondering, "who was hiding a rabid gorilla in Becky's dorm room?" We need more information. We could make this a complete sentence by saying, "Jennifer was hiding a rabid gorilla in Becky's dorm room."
Another fragment: The Skidmore student.
A complete sentence needs a "who" and a "what." The sentence above has a subject ("who") but no verb ("what"). So, the sentence leaves you wondering "what about the Skidmore student?" As a complete sentence, this might read, "The Skidmore student fell asleep in class."