Banks are special consoles that can be found in each of the three wings (Meta/Meta, Tech/Tech and Magic/Magic) of the Watchtower and the Hall of Doom.
They all give access to the same extra stocking place for each player; who can put items there and retrieve them by interacting with any of the console later.
The number of cells available for stocking is not fixed, it can by expanded by buying extra spaces in the Market Place in a similar way to the Inventory extra spaces.
Normally, free players have 12 bank slots, Premium 24 and Legendary 48. There are also 9 "shared" bank slots available to all characters of that account.
JLA Watchtower Bank
Hall of Doom Bank