Some items in the game are flagged as unique. You can only have one copy of a unique item in all of your inventory, including what you have in the bank. If you have an opportunity to acquire another of the same item then you will be unable to. For example, if a mob drops the same item that you already possess, you won't be able to loot the item. If the mob is skinnable this will also prevent you from skinning the mob.
This can be particularly troublesome with unique bag items since you want to have over a dozen bags total preferably. If you are upgrading a bag, and have a choice of upgrading a unique or a non-unique with the same number of slots, replacing the unique bag gives you one less potential problem.
It is possible to have someone mail you a second identical item but you will be unable to retrieve it from your mail until you get rid of your previous unique item since you will be unable to collect another identical item. The mail "trick" simply gives another person control of the unique so you can't use it.
To remove a unique item from your mailbox, if you already possess one in your inventory, you must sell the item in your possession to a vendor and then remove the second unique item from the mailbox. You then dispose of it appropriately (such as mailing to another character) and use the "buyback" feature to return the original item to your inventory.
A handful of items are "Unique (X)", where you can only have X copies of that item between your inventory and bank. For example, if you have five of an item with the "Unique (5)" property, you cannot loot or otherwise acquire any more of them.
If an item says it is unique-equipped, it means that you can have only one of those items equipped at any one time. You can have as many as you see fit in your inventory, but only one actually equipped to one of your slots. Some particular items (namely Jewelcrafting-specific gems) permit a specific number (greater than one) to be active at one time.
This is mostly to prevent overpowering of characters with powerful gems, but still allows them to craft and sell them to others even when they already have one. It is also used on rings and trinkets, especially epic ones, to prevent you from equipping two of them while still allowing you to loot more than one, presumably to sell or disenchant.