A bank is a financial institution whose primary activity is to act as a payment agent for customers and to borrow and lend money. The first modern bank was founded in Italy in Genoa in 1406 its name was Banco di San Giorgio (Bank of St. George). They are important players in financial markets and they offer financial services such as investments funds.
There are several bank types. First type is the retail banking which deals directly with individuals and small businesses. Business banking is also other activity that provides services to mid-market business. Corporate banking is directed at large business entities. Private banking is the bank providing wealth management services to high net worth individuals and families.
Last type is the investment banking that is related to activities on the financial markets.
Most banks are profit-making, private enterprises. However, some are owned by government, or are non-profits. Commercial bank is a type of financial intermediary that is used for a normal bank to distinguish it from an investment bank.
Commercial banking is also known as t business banking. The U.S. Congress after the Great Depression required banks to only engage in banking activities, whereas investment banks were limited to capital market activities. Since the two no longer have to be under separate ownership, some use the term "commercial bank" to refer to a bank or a division of a bank that mostly deals with deposits and loans from corporations or large businesses.
Private Banks are banks that are not incorporated. A non-incorporated bank is owned by either an individual or a general partner(s) with limited partner(s). In any such case, the creditors can look to both the "entirety of the bank's assets" as well as the entirety of the sole-proprietor's/general-partners' assets. Private banking refers to non-government owned banks in general.