In Civilization and Civilization II, a bank is a city improvement related to tax and luxury rates. A bank requires the advance of banking and a bit of production to create. It also requires a marketplace in the city already. The bonus it creates is a %50 bonus to luxury and tax.
A bank can be a very important wonder, especially under republic or democracy. The reason for this is that republic and democracy can produce a great amount of trade, but also a great amount of discontent. A bank turns a large chunk of that trade into luxury, which eases the discontent.
The effect can be quite noticable, in that building a bank can change a city on the edge of unrest into a city celebrating "We Love the President Day".