The White String Thing.........

Can anybody tell me what that white stringy thing is that seems to be attached to the yolk when you crack open an egg? And - are you really supposed to eat it? What is it's purpose - is it part of the egg white - or the yolk? I never realized that a chick had an umbilicle cord! Are chickens now mammals? (Yes - I'd like a glass of chicken milk please!).........Just a dizzy question..........DIZ