After trying to explain this method of rendering beeswax to some friends, I decided it would be a better idea to do a step-by-step post with pictures.
I have tried rendering beeswax several different ways, and this is by far the easiest and least messy way I have found.
You can use burr comb...
...or cappings from your honey comb, or a combination of both.
You will need:
Burr comb and/or wax cappings (put the cappings outside for a few days first, and let the bees clean off as much of the honey as they want)
A disposable container or a container you will use only for beeswax*
A cut-away lid(more on that later) or rubber bands
A paper towel
A Styrofoam cooler
A piece of glass large enough to cover the cooler
A warm, sunny day
*It is possible to clean the beeswax from the container, but it's a lot simpler to not have to.
Line the bottom of the cooler with aluminum foil
Put an inch or two of water in the bottom of the container. This makes it easier to get the hardened wax from the container later on, since the wax will float on top of the water instead of sticking to the bottom.
I like to use a plastic container with a snap on lid because I can cut out the center of the lid and still use the rim.
Stretch a paper towel completely over the mouth of the container and fasten it with a cut out lid or rubber bands. (The rubber bands will get hot and break, so it's a good idea to use more than one.)
Wad your wax into a ball...
...and place it on top of the paper towel
Place your Styrofoam cooler in a warm, sunny place. I use the roof because it gets warm really fast.
Place your container of water with the beeswax on top into the cooler...
...and cover everything with a sheet of glass.
The wax will melt at around 147*. Keep an eye on the temperature as it melts - the wax will become discolored if it gets too hot. Ideally, you don't want it to get above 150*.
(Beeswax is also flammable if the temperature gets too high.)
When finished, it will look like this.
All the wax has melted and dripped through the paper towel into the container below, while the paper towel has caught the impurities. (Sometimes it requires a second or third melting/straining to get out all the impurities - it probably depends more on the brand/thickness of the paper towels than anything. If your wax still has dark or dirty specks in it after one melting, let the wax cool, break it into pieces, change the water and the paper towel, and repeat the melting process.)
This dark substance on top of the paper towel is called "slum gum". Spread it out thinly and let it dry completely, then break it into chunks and use it as fire starter in your smoker.
When you remove the paper towel, your clean wax will be floating on the water below.
If you cool the wax too quickly, it will look like this.
If you care about the way your wax looks (I didn't, because I was only going to break it into chunks to melt down for lip balm) you will need to cool it gradually. You can do this by propping the glass up a little and letting some air in. If you don't move the container at all until it is completely cooled, you should end up with a nice, smooth, piece of beeswax.
The wax can then be re-melted and used for lip balm, candles, lotions, or whatever else you want to use it for.