The banker is a moneymaking NPC who spawns automatically once you have the “church” built from upgrading to a tier 4 kingdom. He goes inside during the night, making him and all your money that he holds completely safe from greed. During the day you can give him coins, which he will collect and store in the bank. He can only carry 10 coins at a time, while the bank can hold an infinite amount of coins.
For each day that passes, the amount of coins in the bank increases by 7% (it can never exceed 8 coins per day, so at 101 coins your interest stops growing). The interest is rounded up, so you will get at least 1 coin per day, as long as you have invested at least 3 coins.
Because of this, it is a great idea in the long run to store coins with the banker. During the day, if you have more than two coins in the bank, you can withdraw your stored coins by walking by the banker. The amount you can withdraw at a time is either 33.3% (1/3) of your stored coins, or the amount needed to completely fill your coin pouch, whichever is lower.
He never gives you more than your bag can hold, but tends to disregard gems, so he might give you too many coins if you are carrying some gems.