As you can see from the diagram below, the human nervous system is divided into two main components, the peripheral nervous system and the central nervous system.
The peripheral nervous system is further divided into two components. The somatic nervous system involves the voluntary movement of skeletal muscle. When you decide to raise your cup of coffee to your mouth, that is the somatic nervous system at work activating the appropriate muscles. The second component of the PNS is the autonomic nervous system, which controls processes that are out of our conscious control.
The autonomic nervous system is further subdivided into the parasympathetic nervous system and the sympathetic nervous system (which you are interested in). In very generalized terms, they have antagonistic effects on each other. The sympathetic nervous system generally activates the body, whereas the parasympathetic nervous system primarily acts to relax the body. This is a simplified view of things, but it is useful to think of it in this way.
The sympathetic division of the autonomic nervous system increases heart rate, inhibits digestion, relaxes bronchi in the the lungs to allow more airflow, causes the release of glucose from the liver, and relaxes the bladder. All of this activates the body, prepares it for a "fight".
In contrast, the parasympathetic division reduces heart rate, constricts bronchi in the lungs, and increases blood flow to the digestive tract. It has been called the "rest and digest" system.
As you can see from the diagram, the nerves of the parasympathetic nervous system come directly from the brainstem and spinal cord. However, with the sympathetic nervous system, there are "ganglia" (groups of cell bodies) outside the spinal cord itself.
Interestingly, the parasympathetic nervous system only uses Acetylcholine (ACh) to exert its effects on organs, but the sympathetic nervous system uses both ACh and norepinephrine (NE) to activate the body.
Its important to remember that the sympathetic and parasympathetic nervous systems are autonomic, meaning we generally have little control over them.