Your dwarves will bring mostly animal items to a refuse stockpile, such as chunks, remains, bones, shells, skulls, corpses, raw hides and any rotten items. This helps in avoiding miasma from rotting items in your fortress, if the stockpile is placed right.
A refuse stockpile, or rather, its placing and potential content, poses a surprisingly big challenge. Placed subterranean it will cause fast spreading miasma, obscuring view and giving dwarves unhappy thoughts. Placed above ground, where no miasma is created, it usually forces your dwarves to walk into danger, especially during a siege if they try to collect corpses(The latter can be fixed by changing the "forbid" default settings). Additionally you may want to prevent certain contents like bones from disappearing (this happens above ground), a cause for additional micromanagement.
It is useful to keep a refuse stockpile in an enclosed room next to your butcher's shop to store corpses waiting to be slaughtered, prevent clutter, and lock in miasma from rotting butcher leftovers. Butcher's shops tend to become cluttered very quickly, as a single animal already "bursts" into many items.
It is also useful to keep a subterranean refuse stockpile holding only shells, skulls and bones near your craftsdwarf's shop for bone carving and bowyer's shop for making bone crossbows. Such a stockpile will not create miasma because bones and skulls never cause miasma (above ground they will disappear, stockpiled or not).
You can provide your stockpile with 2 doors arranged as an airlock. This will mostly prevent miasma from spreading when a dwarf steps through. You can contain miasma completely by building a diagonal entrance to the refuse area, since miasma cannot travel diagonally.
Corpses, chunks, body parts, skin, and food all can rot, emitting miasma when stored on Dark Subterranean tiles. It is important to know where you want to place your refuse stockpile beforehand, as miasma causes an unhappy thought. Designating a refuse stockpile on any tile with Light Above Ground will prevent miasma, though any items, including bones, shells, and skulls in such areas will occasionally be removed from the game.
Alternatively, you can dig an open pit down to the z-level most convenient for you. Light need only touch the tile once for it to become permanently Light Above Ground, after which you can rebuild the roof.