Basics
The dungeon disenchanting feature was introduced to the game with patch 3.3. As long as a character with the Enchanting profession remains in the group and has a high enough enchanting skill, all group members will have an additional "Disenchant" option available when rolling for an item. This option works exactly like the "Greed" option and has the same priority; however, if you select "Disenchant" and win the roll, the item will be automatically disenchanted and you will receive the resulting materials rather than the item.
Yield
Disenchanted items yield different materials depending on their quality and level. Items of uncommon quality may disenchant into either dust or essences of various types, with armor having a higher chance to yield dust and weapons having a higher chance to yield essences. Items of rare quality disenchant into one or more shards, while items of epic quality disenchant into shards if they have an item level below 56 and into crystals if their item level is 56 or above. Items of legendary quality may not be disenchanted. Occasionally, disenchanting an item will yield a material that would normally result from disenchanting a higher-quality item. For example, disenchanting an item of uncommon quality will very rarely result in a shard.
Benefits
Before this feature was introduced, only characters with the Enchanting profession could gather Enchanting materials from items found in dungeons. This resulted in the materials being scarce and expensive to purchase from the auction house and proved quite frustrating for most players, since an item may sell to a vendor for only a few gold, but disenchant into materials worth several hundreds. Following the change, Enchanting materials have become less scarce and expensive, allowing players to more easily obtain enchantments for their gear.
Considerations
Familiarize yourself with the various types of Enchanting materials and their current value. Although most items are worth more when disenchanted than when sold to a vendor, items that disenchant into extremely common materials may be an exception. For example, if a weapon is worth 20 gold when sold to a vendor and disenchants into a shard worth 11 gold, choosing the "Disenchant" option would be a gold loss. Similarly, certain items may be especially valuable when sold through the auction house, fetching prices several times higher than what the enchanting materials resulting from them would be worth.