A bank is a building with several teller NPCs that allow a character to access a private storage area that is accessible from any bank branch. Branches appear in major cities and some towns, and allow a character to store items not currently in use.
All of the banks share the same vault; the neutral banks are equivalent to the faction banks. 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; ATM's for short.) However, players are not able to use the banks of opposing factions, because they cannot talk to the tellers.
Initially a character's bank account (vault storage space) contains 28 item storage slots at no cost. An additional seven bag slots for more space are purchasable. These provide no additional item storage slots, but allow placement of a bag that has item storage capacity. The bags can be changed out and it is common practice to progressively place larger bags. The cost of the bag slots increases with each extra slot purchased starting at 10, (then increasing to 1, 10, 25, 25, 25, 25, for a total of 11110 for all slots (not including the bags that go into them).
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: