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
Its not always obvious who needs an item, perhaps the Mage already has better armor or perhaps the party loots some crafting items and its 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. But if one or more people declare need only they roll on the item.
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.
Off Specs
As Hybrid classes (Druids, Shamans, and Paladins) and other classes capable of performing multiple roles (Warrior, Death Knight) classes, items may drop that may be used for any of these roles. Many players have different perspectives on rolling for off spec loot. Some players expect others to not roll on an item that is not for their current role. For example, a Retribution Paladin or Fury Warrior are expected to pass on a piece of tanking plate, and if the tank does not need that item, then they should either pass or roll greed.
Problems with Need before Greed Loot Level
The Need before Greed Loot Level takes into account whether or not you can 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 will not be able to roll on it. Similarly, a level 39 hunter will not be able to roll on mail items even though they will be able to use it next level.