At its most basic level, the nervous system, in the human body, is the communication system between the entire body and the brain. It is the system which carries signals from the brain to the rest of the body, and back. These signals are either voluntary or involuntary, and result in an action, such as movement, breathing, or the beating of the heart to make the blood flow.
The nervous system is made up of two parts, the central nervous system (CNS) and the peripheral nervous system. The CNS is the control center. It is made up of the brain and spinal cord. This is where the "messages" originate. If part or all of the CNS shut down, part or all of the rest of the body shuts down.
The PNS is made up of neurons, neurotransmitters, and nerves themselves, which act as conduits for messages from the brain, telling the body exactly how to function. The PNS is broken down into several smaller categories of systems designated by what kind of messages they transmit, including voluntary messages (such as large muscle movement) and involuntary messages (such as the heart beating or the gastrointestinal system digesting).
The links below contain more detailed information about the CNS and the PNS.