Century? Thousand years old? More like 90 day old preserved eggs. These eggs are used as a garnish or ingredient, or just eaten cooked. They are eggs preserved in sodium carbonate and clay, a chemical reaction causes the colour change. The taste is very similar to an unpreserved egg, but the colours and look make them a spectacular show food.
In the photograph you can see a duck egg in it's shell at the top right, below that a peeled egg , and the eggs sliced open on the left. Notice that the preserving process makes the egg solidify.
Another variation of these eggs is the ones packed in green tea to give them an extra green tea flavour. As you can see the shell turns green, but once peeled they look very similar.