Inboxes contain all Letters and Parcels that are mailed to a character in a town and items purchased from the Market.
Part of this change is that items in the Inbox do not count towards the item limit in the Depot Chest so these items in the Inbox cannot be moved inside the Inbox and no items can be added to the Inbox from the Depot Chest, except by parceling.
Prior to the Autumn Patch 2012 items located in an Inbox were unique to that city, just like all other items stored in a Locker. For example, items that are in your Carlin Inbox will not be available to you in your Thais Inbox, and vice-versa.
There is not a maximum amount of items that can be held in an Inbox, since it seems they are being added into the game to combat destructive behaviour like "letter or parcel bombing".
With the Autumn Patch 2012 all Inboxes have been merged into one Inbox to be made globally accessible: items in a character's Inbox can be accessed by that character from any city. Therefore, it is not necessary to specify a city when mailing letters or parcels.