What is the white stringy stuff inside of an egg when you crack it?

The strands are the chalazae which anchor the yolk in the center of the thick white.

They are neither imperfections nor beginning embryos. The more prominent the chalazae, the fresher the egg.

Chalazae do not interfere with the cooking or beating of the white and need not be removed, although some cooks like to strain them from stirred custard.

Source: Frequently asked Questions about Eggs, http://whatscookingamerica.net/Eggs/FAQ.htm