The on going attack is in direct result to Godaddy's support for Protect IP, SOPA, and their support for ACTA. The attacks originally started though, when their CEO celebrated killing an elephant for sport & posted photos and video of this as bragging rights.
At this time Anonymous hasn't attacked the customer side of things as it is greatly believed that the unknowing and innocent customers shouldn't be forced to suffer for the poor actions of Godaddy.
So the attacks have been directly on Godaddy.com and the back-end to prevent new customers from signing up. Sometimes even being re-directed to other domain registers.