A purse can be cut and a corpse can be looted, and everything you were carrying when you fell belongs to whoever reaches you first. A bank account is the one place in Skyrim where that is not so. We are starting with three houses, one in Windhelm, one in Markarth and one in Solitude, and an account opened at any of them is an account at all three.
How it works
An account costs 100 gold and you open it in person, standing at a teller's counter. The fee is spent rather than held for you, so opening an account is a decision rather than a formality.
The three counters are the East Empire Company warehouse in Windhelm, the Treasury House in Markarth, and the East Empire Company docks in Solitude. You have one balance and all three read it, so gold deposited in Windhelm is gold you can draw in Markarth.
Reading your balance works from anywhere. Everything that moves gold needs you at a counter, and the refusal will say so plainly: no clerk can hear you from across the province.
Banked gold survives death. When your body is looted and your purse is gone, your account is exactly where you left it (see Death Has Weight).
The vault is the second half of the house. It is storage for items rather than coin, opened from the same counter. It is built for ordinary and static gear, so keep crafted potions and self-enchanted pieces in a chest you own until the item work behind the vault is finished.
Commands (/help company)
/bank shows your balance, or the prompt to open an account, and names the three branches./bank open opens an account at the counter you are standing at. 100 gold./bank balance reads your balance from anywhere./bank deposit <amount> moves gold from your purse into the bank, at a counter./bank withdraw <amount> moves gold from the bank into your purse, at a counter./bank vault opens your item vault, at a counter.
Group gold, shop stock and faction vaults are a different system (see Treasuries, Vaults, and Shops).
