A coregent is a dual kingship.  This was common in ancient times and even in modern day monarchies and dictatorships.  When a king became old and feeble his oldest son would often take over the throne.  The son would have the authority and power of king but his father still was king as well and he held authority. 

A modern day example of a coregent is found in Cuba.  When Fidel Castro became ill, his brother Raul took the presidency with his brother's authority.  While Raul is president, Fidel still remains in the picture and his wishes also need to be taken into consideration.

A coregency sometimes makes biblical timelines difficult.  Often the reigns of kings overlap and it is hard to determine when one kingship ends and another begins. 

In Daniel 5 King Belshazzar sees a hand appear and write a message on the wall.  He offers the third highest position in all of the land to the person who could translate the message.  Why the third highest?  Because even though he was king, he was really only second highest.  His father was still living but was in bad health and was away at a hot spring.  He was a coregent.