"Double boiling" is a very simple idea, based on the fact that water boils at 100 degrees celsius. If you place food in a pot, and then place that pot in boiling water, you're cooking the food at 100 degrees.
Why might you care about that? Well, it turns out that a lot of very tasty sauces and reductions are pretty sensitive to temperature, and you don't want to burn them or dry them out. See, for example, the chocolate sauce I've linked.
More generally, you can use this technique whenever you want to heat something in a pot (especially liquids) but you want to know that they won't get too hot.
So, how do we do it? Well, take a large pan or pot and put in about a 1-2 cm layer of water; heat it until it boils, and then put in a second smaller container inside. A small coffee pot is ideal, but a pyrex bowl can just as easily do the job.
Alternatively, you might run into a device called a "double boiler"; this is just a specialized pot with two layers, one which holds the boiling water and one which contains the thing to be heated. It's hardly necessary, but it can be a useful shortcut.
Note that the second bowl will be in direct contact with a rather hot surface, so if you're concerned about it burning / cracking, then you can prop it up on some supports (e.g. chopsticks, a couple of chopped carrots, or anything else you don't care about heating aggressively).

