The short answer is that chemically, chocolate is composed of lots of different little crystals (six to be exact) but the desirable ones are called beta crystals. The development and formation of these beta crystals are what makes well-tempered chocolate.
If the cocoa butter rises to the surface, some people commonly think their chocolate's gotten moldy and toss it out. If you've tossed out your chocolate, you've tossed out perfectly good, but unattractive, chocolate.
If you made chicken stock and refrigerated it, would you toss it out when the fat collected on the surface? If you made vinaigrette and the olive oil rose to the surface, would you throw it away?
To avoid the dull white sheen on the surface of the chocolate.
That's what happens to chocolate that's not properly tempered: the cocoa fat rises to the surface and "blooms", making it unappealing and unattractive. When you buy chocolate, like a candy bar, the chocolate's been tempered and it should be nice and shiny and snap when you break it. If you leave your candy bar in a warm car and later open it up, often it'll become white and grey. The heat caused your chocolate to lose its temper. When you buy chocolate for baking, it should arrive well-tempered. But once you chop it up and melt it, the beta crystals change, the chocolate loses its temper, and you'll need to re-temper it again if you plan to use it as a coating. If you're going to cook with it, just use it in your recipe, as indicated.
Pages and volumes of technical research have been written about tempering chocolate, but here are the main reasons for all you home cooks out there:
You don't need to temper chocolate if you're going to bake a chocolate cake or make chocolate ice cream. The only time you need to temper chocolate is when you need an attractive, shiny coating for candies that will sit at room temperature. You can get around tempering by dipping chocolates in melted, untempered chocolate and storing them in the refrigerator. Just remove them from the refrigerator a few minutes prior to serving them. The coolness of the refrigerator will stratify the cocoa fat.
There are many different methods for tempering chocolate. Some are really complicated, and some are really messy, especially for home cooks.
Many professional pastry chefs and chocolatiers can instinctively tell when chocolate is perfectly tempered by looking at it or touching a smidge of it to their lip. A thermometer is foolproof.
There is a simple 3-step method that's a snap for home cooks. All you need is an accurate chocolate thermometer (although a good digital thermometer will work).
A tempering machine can do all of this for you, but it's not as much fun ;)