As the name implies, a deposit box allows the player to deposit items into their bank account, which can be done without the entry of the player's bank PIN. However, players cannot use the bank deposit boxes to withdraw items and they also cannot check their bank content using the bank deposit box. They were added at the same time as bank PINs, in September 2005.
To use the deposit box, players simply click the box to open up the bank deposit box window, which is an interface that shows the same items which were in their inventory before clicking the bank deposit box. They can then deposit items by clicking on the item, and can choose to deposit more than one as if they were accessing their bank account through the normal method.
Players can also deposit items to the bank deposit box without opening the deposit box window. To do this, players click on the item and choose "use item" and then clicking on the bank deposit box. A window will open in the player's chat screen, asking the number of items the player wanted to store into their bank.
Again, once the items have been deposited, it is impossible to get them back other than through normal bank account access, which will require the player's PIN (if any). Presumably, Jagex added them so that players locked out of their banks due to the PIN system were still able to play. A player who has forgotten their PIN can cancel it and use a deposit box to store items during the 3 to 7 day cancellation period.
This list is incomplete.