The song originates from the IRA Border campaign of the 1950's. The "hare" is an IRA volunteer, Barney Conway is a paid informer of the British and the "Pedigree greyhounds" are British soldiers who tried - and failed - to capture the "hare".

It's written in the Dark Rosaleen or Kathleen Mavourneen style of song where Ireland is portrayed as something other than the country.

Hope that helps!