Candling is the process of shining a bright light into an egg to have a look at the contents. It used to be done with a candle in a dark room, but now high intensity light can be used in a darkened box. For instance, here is Brinsea's egg candler:
Commercially, eggs are candled to make sure they are free of blood or meat spots. This is why, incidentally, white eggs used to be the most popular type of commercial egg. It's not just a matter of finding a high production, breed, but also of easy examination to make sure the contents are free of unwanted matter. Now, more high production breeds can lay brown eggs, and we have better candling devices.
You might also candle eggs when you're incubating them, to monitor the development of the embryo. Or if yo've had eggs shipped you can candle before incubation to check on the condition of the air cell.