"Increases the amount of gold generated by each colonists and surrounding dwellings."
In order to build your first Bank, you need to reach a colony population of 400 colonists. The second Bank needs 800 colonists, and the third needs 1,200.
The Bank increases gold gain from all Colonist buildings in a range for 12 blocks (making it a 27x27 size square) by 30%.
Similarly to Markets, Banks' buff areas of effect do NOT stack with each other for the same Colonists buildings, and you cannot have more than 3 Banks simultaneously (unless electing Mister B as Mayor, which provides a free Bank that can be built as your 4th Bank).
Note that Bank itself doesn't need gold to maintain.
There are many different theories as to the optimal layout of housing around a bank. See the discussion of housing layouts.
As of V0.5.0, there is a notable anomaly though with these zones icon showing: Housings which are only half build inside the zone do show the icon on the upper part of the Market(so top-left and top-right), but don't show it at the bottom. However, the building has to be inside the range completely in order to get the bonus, even if the icon is showing, the housings built only half-part-inside don't get the bonus.