The Bank of Varrock
You can only hold 30 items at once in RuneScape and carrying around all your money is risky, because when you die you drop what you are holding. Therefore you need to somewhere to store your extra items and to stash your treasure.
For this purpose the bank of Varrock has opened branches all over RuneScape. Branches of this bank can be found in Varrock, Falador, Edgeville, and Draynor village. When you wish to access your in-game bank account just walk to the nearest bank and speak with one of the banker characters inside.
The bank screen shown above will appear, and you can now choose to deposit and withdraw items. All the items in your account, as well as the items in your inventory will be shown. The bankers aren't fussy and will safely store anything from runes to big piles of coal!
Each item has 2 numbers next to it. The green number indicates the number in your bank account, and the blue number indicates the number you are holding. Click on an item to select it. You can then use the deposit and withdraw buttons at the bottom of the screen to move it in and out of your bank account. To deposit a single item press the button labelled "One", to deposit a large stack of items just use the larger numbers. (E.g in the example above pressing the "100" button would deposit 100 coins into the bank account)
The banks are free to use, so making use of them to safely store your extra items is a very good idea!