Burning vs Melting - when applied to cooking chocolate I'm aware melting is simply heat causing molecules to go from solid to liquid, and burning is a chemical reaction, normally with oxygen; but I don't understand how this applies to cooking chocolate.
See, when you cook chocolate naked on a pan, the chocolate will burn. However, if you cook a bowl of water with a smaller bowl floating on top with the chocolate in, the chocolate will melt and not burn. So, what's going on here? FYI, I'm not a physics major.
 A: When you use a bare pan it conducts heat from the burner and the metal will reach a very high temperature; enough to carmelize, and possibly burn the chocolate.
But when you put water in the pan, bring it to a boil the water basically regulates its temperature at phase transition (liquid to gas) to the boiling point, about 100 deg C. So the bowl, in contact with the water cannot exceed 100 deg C as long as there is water in the pan and the bowl does not contact the metal of the pan. At 100 deg chocolate simply melts.
A: The inconsistancy you see is due to different use of the term "burn" in a non-technical setting.
In cooking, it is valid to say "The chocolate burned to the bottom of the pot".  That means you now have a sticky sludge stuck to the bottom of the pot that is hard to clean up.  It does not mean the chocolate underwent a exothermic oxidation reaction, which "burn" as you're thinking means.
Words can have different meanings, and the setting and context is important.  Especially old and common words used in every-day non-technical contexts will often have colloquial and context-specific meanings.
If you sit on the beach for a while you may get a sun "burn", but that isn't really a chemical oxydizing reaction either.  A freezer "burn" isn't.  I cringe when I hear that expression too, and refuse to use it myself.  But, it's common enough and actually refers to freeze-drying.
By the way, you're "burnt" chocolate probably had the sugar in it carmalized.  There are likely other chemical changes that are only re-arranging the atoms and molecules that were there already.  Another common example is frying eggs.  Eggs clearly undergo a change, but not really a reaction with some external substance.
Keep in mind that proteins are chains of amino acids, but the properties of those proteins come mostly from the shape of the balls those chains wrap themselves into, not the makeup of the links.  Heat shakes up the ball enough so that sometimes it finds a lower energy state.  Sometimes these balls can then bind to other balls in different ways.  Runny and clear egg white turning milky and much more solid when heated is a great example.
A: Olin Lathrop captured something important, which is that the word "burn" does not always align to the chemistry definition of the word.  But I did want to answer your question.
Chocolate will burn in a "naked" pan because the heat is not evenly spread.  In cooking, it's desirable to have the top surface of the pan have an even temperature across the entire cooking surface.  However, in practice, that can be difficult to achieve.  Stoves often provide very uneven heating to the bottom of the pan, especially gas ranges which may only heat a circle!  To spread the heat evenly, a pan must be very thick, and that is costly.  Pan manufacturers try to strike a balance between a pan that's thick enough to spread the heat but thin enough to be affordable.  High end pans will often have several layers of metal on the bottom to achieve a more even spread.
Chocolate is particularly demanding regarding temperatures.  The science of chocolate is amazing, and I highly recommend looking into it some time.  Chocolateers have to work with 5 different crystalline forms of chocolate that can form when it cools, of which one of them is "the right one."  Accordingly, you must have a very even temperature within your pan of chocolate.  Most pans just can't meet those criteria.
The "double boiler" solves this by heating water and then using that water to heat the container with the chocolate.  First off, you can create a much larger thickness of material between the burner and your precious chocolate.  Instead of having maybe a quarter inch of metal protecting the chocolate, you can have 2 inches of water or more!  Second, water is a liquid so it can move.  The water will form convection cells which are very effective at spreading heat out uniformly throughout the liquid.
Just be careful with the steam!  Chocolate is quite hydrophilic, meaning it is attracted to water.  If any of the steam condenses into your chocolate, the chocolate will immediately absorb the water, which causes the chocolate to "seize" into a useless lump which cannot be used unless you heat the chocolate hot enough to drive out all the water.  This temperature is high enough that it will mess up all of the crystals you were trying to form, and you will have to re-temper the chocolate again!
