Need Before Greed describes a rule of looting etiquette as well as the loot setting enforcing that etiquitte.
While part of a party most players will expect that if they need an item they should get first pick at it before those who want it for Greed. For example, a party consisting of a Rogue, a Mage, and a Warrior walk into a bar and loot some rare cloth armor. All of them can potentially wear that armor, but only the mage really needs it, the rest will likely just sell it or give it to an alt (see Greed). Therefore, it is expected that the warrior and rogue will allow the mage to have it by not rolling on the item.
Etiquette[]
It's not always obvious who needs an item, perhaps the mage already has better armor or perhaps the party loots some crafting items and it's not known who has what professions. Prior to patch 1.9, people would say "n" or "g" to indicate if they want an item for need or for greed. Now this declaration of need or greed is built into rolling for an item. If everyone declares greed, then everyone rolls for the item; if one or more people declare need, only they roll on the item, which is the usual tactic of ninjas.
Loot Level[]
There is a Need before Greed party loot level which is an attempt to enforce the above etiquette. Those party members who can't use an item (such as mages and mail armor pieces) or who can't use the item most effectively (such as warriors and cloth armor pieces) aren't given the chance to loot an item until the party members who can most effectively use the item have the chance to loot it.
Problems with Need before Greed Loot Level[]
Before patch 3.3.0, the Need before Greed Loot Level only took into account whether or not you could use the item now, not in the future. This means that if a mob drops a piece of mail requiring level 37, the level 36 warrior in the party would not be able to roll on it. Similarly, a level 39 hunter would not be able to roll on mail items even though they would be able to use it next level.
At patch 3.3.0 the system was changed so Need Before Greed will now recognize gear appropriate for a class in three ways: the class must be able to equip the item, pure melee will be unable to roll on spell power items, and classes are limited to their dominant armor type (ex. paladins for plate). All items will still be available via Greed rolls as well as the new Disenchant option should no member be able to use the item. Also, players will be able to roll on items with a required minimum level higher than then player's current level.
The changes at patch 3.3.0 deliberately introduced a new complaint: classes may not roll on gear of lesser armor level even if it is an upgrade, such as a holy paladin wanting mail with spellpower and intellect. The only way for that player to be sure of getting that item is for either for all other players to pass, or for whoever wins that item to trade it afterwards provided it wasn't disenchanted. Despite the griping, this is important to ensure equality with other classes: mail wearers can't roll need on plate, so plate wearers can't roll need on mail. Blizzard expects people should be able to find all the gear they need in their own armor class without poaching from other armor classes.
Off Specs[]
As Hybrid classes (such as Druids, Shamans, and Paladins) and other classes capable of performing multiple roles (such as Warriors and Death Knights), items may drop that may be used for any of these roles; NOTE: This may not apply to Priests because they may or may not use the same gear for both healing and DPS. Many players have different perspectives on rolling for off spec loot. Many players agree that gear rolls should be main spec only, which in most players' opinion, is the role that they are currently playing with, even if it is the player's Secondary spec. In a raid setting, it would be more beneficial to the raid if the active tank is allowed to roll for a tanking item over a Paladin or Warrior currently acting as a DPS.
If you are running an instance or raid with the intention of getting off-spec gear, or there's a chance that off-spec gear will drop, it's best to declare this before the run starts and make sure everyone is okay with it. To be absolutely polite you should be willing to forgo main spec Need rolls.