Varroa mites feed on the internal fluids of honeybees.  Like other parasites in nature, they can weaken and vector diseases and can kill the hive off in less than a year.  These mites travel on honeybees and can spread diseases like deformed wing virus between bees and hives.  Most of the world honeybee populations are adversely affected by this parasite.  Hawaii, in 2007, was the last area in the US to become infected.

Great photo of a Varroa Destructor on the back of a honey bee.  Be sue to click the center of the image to see everything in detail.  

Above are honeybees with many varroa destructors in tow.  Below is the lifecycle of a varroa.  They literally feed on the honeybee pupa during its development (step four) and emerge as the new honeybee does to live to infect other pupa.  After some time, the varroa simply over runs the hive.  Stop the pupa food supply and you can stop the varroa.  Splitting hives, giving the hive some time without egg/pupa productions can break this cycle naturally.

Image from Huang, 2012.