A bank is a building with several teller NPCs that allow a character to access a private storage area (like a large Backpack) that is accessible from any bank branch. Initially it contains 24 storage slots but additional bag slots for more space are purchasable for an amount that increases with each extra slot purchased starting at 10, (then increasing to 1, 10, 25, 25, 25, 25).
The Vault of Ironforge
All of the banks share the same vault. If you place items in one bank, you will be able to access them in another (officially, this is accomplished through the use of Automated Teleportation Modules, or for short ATM's). However, players are not able to use the banks of opposing factions.
Unlike a real bank, you cannot store currency within the bank. Only items.
Where can I find Bank branches?
Faction Bank Branches
All of the capital Alliance and Horde cities have bank branches:
Horde Bank Branches
Alliance Bank Branches
Neutral Bank Branches
Some Neutral cities have them as well: