When your bags are full, the bank holds the rest.
Finding it
The banker keeps the bank, one of the market houses in town. Walk into its door once to open it and again to go in, then right click the banker, or hold your thumb on him, and choose Bank.
Drawers
The bank is a set of drawers, and items take up the same room in a drawer as they do in your bag. Your first drawer is free, and it is there the first time you talk to the banker. More drawers can be bought one at a time, and a locked drawer shows what it costs.
What goes in
The bank holds items, and your gold stays with you. Weight does not count in a drawer, so the bank is the place for heavy things, and an item comes back out exactly as it went in.
Moving things
The bank opens with your bag beside it. Drag something across to put it in, drag it back to take it out, or move things around inside the bank. You can move part of a stack or split one. On a phone it shows one side at a time, with a switch between the two.
It stays put
What is in the bank stays there whatever happens to you or to what you are carrying. A move in or out either completes or does not happen, so an item cannot be lost part way.
Next: gold and the bank, for where the money comes from.