A garbage disposal is any convenient mechanism to quickly (and, preferably, automatically) dispose of the ever-growing mounds of trash produced by a successful fortress. Trash includes vermin remains, spoiled food, excessively worn clothing and goods, stray animal corpses and unusable byproducts, and the remains of invaders who foolishly attack your fortress. Note that, if you choose to dispose of your citizens' corpses in the same manner, you will need to prepare memorials to avoid ghosts and unhappy thoughts.
Properly disposing of trash is necessary to keep your fortress neat, your dwarves happy, and your framerate high. When disposing of clothing and goods, masterwork items will cause an unhappy thought in the minds of their creators. This can be avoided by trading those items to a passing caravan.
 Refuse stockpile
The basic disposal method is a refuse stockpile. Trash placed in a stockpile with refuse enabled will degrade over time, and most items will rot away eventually. A simple refuse stockpile is generally sufficient for a small fortress, though disposal needs grow significantly as fortress size increases. Once your stockpile becomes full, trash will accumulate in your fortress hallways until you resolve the problem. An overtaxed refuse pile is frequently signaled by clouds of miasma.
Combining your refuse stockpile with a quantum stockpile allows your dwarves to stack infinitely many items in a single tile. If that tile is also a refuse stockpile, your trash will continue to degrade and disappear over time. While this design takes care of the problem of refuse stockpile space, it does not reduce the game's item count nearly as quickly as the disposal methods below.
A dwarven atom smasher can pulverize any amount of trash instantly. Initial designs are often manual (using a garbage dump zone and the dump command), however when combined with a quantum stockpile, your dwarves will automatically dump their garbage into a pit for easy disposal. The atom smasher can be operated manually, or automated with a repeater. Be warned that an atom smasher is capable of tossing trash back up several z-levels and injuring dwarves above; for safety, a garbage chute should be employed of two or more z-levels, especially if the atom smasher is automated.
A small pool of magma can conveniently incinerate most garbage, including corpses. This system can be manually operated (via a garbage dump zone) or automated (via a quantum stockpile). It is highly recommended to locate the magma at least three z-levels below the dump location (to avoid deadly magma mist). If you pipe your garbage chute down to the magma sea, even magma-safe items can be disposed of with ease.