How do the sympathetic and parasympathetic systems interact to produce changes in our bodies during emotional?

The two emotional states of the body can be roughly divided into two categories - "fight or flight", and "rest and digest". The sympathetic nervous system is responsible for the fight/flight, when the body is stressed, and the parasympathetic is responsible for the rest/digest.

The two systems exist together, but when one is at a peak, the other  is at its lowest point.  - A way of visualising this is imagining a graph of sin(x) and cos(x)

Therefore, when the sympathetic is at its peak, the body is using its energy to prime itself against a perceived threat (preparing clotting factors, increasing respiration and heart rate). When the parasympathetic is at its peak, the body is using its energy to digest food and relax, promoting the digestive system and dilating blood vessels for relaxed blood flow.