They never actually clearly tell us in the comic or tv series.
Robert Kirkman said in an interview he won't reveal the cause of the outbreak. They touch on it lightly and a corrupted government, but never actually will tell.
However, they work out... the Zombie bites are not how the "infection" is spread. Instead, zombie saliva contact with blood cause infections and are 100% lethal. Characters have speculated, but cannot confirm, that whatever causes zombies to re-animate is some sort of virus that is only triggered by death. It is known, however, that upon death, it can take anytime between three minutes and eight hours for the corpse to reanimate.
They believe it's a bio-engineered virus, maybe originally the rabies virus. The government or someone has been messing around with and it's started spreading via human/animals and/or via a contaminated water supply.
The original cause of the zombies was through the vaccination that took place. There were also other walking dead zombies that evolved from the Romero's spin. This is because any zombie that is dead comes back to life.