The phrase used by George Bush during his campagin to show his more affectionate side. It's like a redneck shooting cans with a paintball gun rather than a rifle to be more affection. To get a better understanding read the story of the lions:
One day, one lion said to another "compassionate predator...i like it"
Compassionate conservative- Americans are hopeful that in Bush's second term, the compassionate side will kick in.