First, a safety moment. DO NOT LET THE POT BOIL OVER!!!. If you have an open flame this is especially dangerous, and having experienced the risk myself, prompted this blog post. I got distracted, the mix started to boil and froth up, and boiled over, causing flames to start leaping up off the pot, just like a kitchen grease fire. Beeswax is a hyrdocarbon, and with readily burn with a good ignition source.
Ok, now for the topic at hand, how to render down beeswax. Now there may be several ways others have done it, but I'm posting what I do, as it is working for me. I've tried several variations and will outline them at the end.
In general, the beeswax I collect is mostly from the cutout jobs that I do. Once a year when I harvest honey I get some high quality cappings wax as well. But I mix it all together. When I do a cutout job, the combs are offered out for open robbing until the bees clean up the honey, then I melt it all down.
First step is setting up the melting pot. Pick a pot that you don't care too much about, because it's going to get nasty beyond repair. Put the pot on your OUTDOOR burner and fill about 1/3 full with water. Fire it up and get it started warming up. Then start adding your beeswax. As you place the combs in, push them down with a stick to help them fill with water and sink. Otherwise they can tend to float on top the water and the heat distribution is disrupted. As the combs start to melt you can keep adding more wax but leave room in your pot for effective stirring. Also, if your wax has a lot of debris (brood, cocoons, etc), you should melt down less per batch.
Try to keep the whole mix from boiling. The wax will melt well below the boiling point of water, and boiling only causes more risk of flash fire, and also can cause discoloration of the wax through over heating. Continue to stir the mix until you have a liquid homogeneous mixture. Now turn it off, or drop your heat down low. If you're wax is mostly clean, you're done. Turn the heat off and let it set up and cool. I pour the mix into a bucket so I can reuse my pan to start another batch. As the mix sits and cools, the wax will float to the top and solidify. Solids and debris will sink to the bottom, leaving you with a nice clean brick of wax floating in the water.
If your wax has a lot of debris in it, you'll want to pre-filter it before letting it settle and set up. Use a screen mesh sieve with a handle, and the flat end of your hive tool as a press. Dip about a half cup of the mix off the surface of the pot, and use your hive tool to press water and molten wax out of the debris, letting it liquids fall back in the pot. Once you feel you've squeezed out as much as possible, dump the strained debris in a bucket to the side. If in the end your pile of refuse sets up in solid block, it still has a good bit of wax in it. If when it cools it is crumbly, you've done a good job. Once you are done with the pre-filtering process where each dip gets you little debris, then you're back to basically "clean" wax. Give a good stir and you are done. Same as before, you can dump into a spare bucket if you need to run another batch, or just leave it in your pot to settle out and solidify.
In the end, your wax may still have some suspended solids in it. The secondary, and ideally final render, is to place the block of wax in an oven on low heat. Set up with a pan of water, then screen to hold the wax above the water, then a paper towel to catch fines. As the wax begins to melt in the oven, it will soak through the paper towel and drip in the pan of water below. The paper towel will catch and retain the fines, and should leave you with a pure clean beeswax.
For casting into molds, take the block of clean pure beeswax, and use a double boiler method to melt it down. I like a glass measuring cup with a spout in a pan of water. Boil the water and stir the wax as it melts. Once it's all liquid, pour it into whatever mold you are using. Let it set up and you're done! If you are making candles in glass jars, sometimes you can get trapped air bubbles. It's a good practice to take the finished candle and heat it up in the oven once more until it melts. This will let air bubbles escape, then the wax will cool and set up again, and you, at last, finally, are DONE!
If you are doing a small batch of clean cappings (honey has been washed off), you can do it all in one run. Set up your final render setup with the screen and paper towels, and just place a handful of cappings on top of the towel. I only use the big pot of water for large batches of relatively dirty wax.
Solar Melters: Solar melters are a great passive way to melt beeswax. I've built one (poorly) and used it with no real satisfaction. When built and used correctly they can be very effective. Look up Ralph Jones III on YouTube or Facebook. He built one out of an old chest freezer that seems to work very well.
I also once tried a sort of inverted water melt system. I filled the pot half full of water, then placed the wax under a bowl shaped screen in the bottom, so it was submerged under the water. The idea was that as the wax warmed and melted, it would float up through the screen leaving the solids all behind, ideally a sort of inverted deep fry where I would get 100% removal of wax, leaving any absorbent debris saturated with only water. Now it did make for a cool lava lamp effect as the wax globules floated up through the water, but ultimately I left a lot of wax in the debris. It just wouldn't come out. Maybe I didn't give it enough time. Also at one point the system boiled, and knocked the screen around so it all got stirred up and mixed anyway.
In the end, I use the multi step system I do, because it works well for my situation. Best of luck, and have fun.