The inside of a chicken's egg is more than just a yolk and the white. In addition to the shell, yolk and a two-part egg white we are all familiar with, there are also several membranes, an air sac, a cord-like connection between the yolk and the two ends of the egg and a blastodisc on the egg yolk, which is where the chick develops if the egg is fertilized and incubated. (The diagram below is copyrighted; you are welcome to link to this page or print it for home use.)
Here's a description of each part idenitfied in the diagram above (which, by the way, is not to scale).
Shell: Made primarily from calcium carbonate, the shell is the last part that's made when the egg is formed inside the chicken. It is somewhat permeable and allows air exchange betweent he egg and the outside world.
Outer Shell Membrane: this membrane is closest to the egg shell interior.
Air Cell: Located in the larger end of the egg, the empty sac is between the two shell membranes.
Inner Shell Membrane: This membrane is between the air sac and the start of the egg white (albumin).
Thin Albumin: The thinner, more watery part of the egg white, furthest from the yolk.
Thick Albumin: The thicker, more gelatinous part of the egg white that is directly around the egg yolk.
Yolk Membrane: The membrane that contains the egg yolk -- when you're making eggs over easy, if you break this, your egg goes runny.
Yolk: The most innner, golden part of the egg, where the chick forms. Nourishment for the growing chick comes from the yolk.
Chalaza: This rope-like white strand connects the egg yolk in the white, you've probably come accross it when trying to seperate eggs as the aprt of the white that clings to the yolk.
Germinal Disc/ Blastodisc: This is the nucleus of the egg, where the sperm enters to fertilize the egg while it is being formed inside the hen.
If the chicken egg is fertilized and incubated, a chick will begin forming inside the egg. It needs to be kept at optimal temperatures for 21 days to hatch a healthy chick. Here are a few pictures of what that looks like. You can see the whole series at the "Birth of a Chicken" post (off-site).