A bank holding company is a company

Holding company

A holding company is a company or firm that owns other companies' outstanding stock. It usually refers to a company which does not produce goods or services itself; rather, its purpose is to own shares of other companies. Holding companies allow the reduction of risk for the owners and can allow...

 that controls one or more banks

Bank

A bank is a financial institution that serves as a financial intermediary. The term "bank" may refer to one of several related types of entities:...

, but does not necessarily engage in banking itself.
In the United States, a bank holding company, as provided by the Bank Holding Company Act of 1956

Bank Holding Company Act of 1956

The Bank Holding Company Act of 1956 is a United States Act of Congress that regulates the actions of bank holding companies.The original law , specified that the Federal Reserve Board of Governors must approve the establishment of a bank holding company, and prohibited bank holding companies...

 ( et seq.), is broadly defined as "any company that has control over a bank". All bank holding companies in the US are required to register with the Board of Governors of the Federal Reserve System

Federal Reserve System

The Federal Reserve System is the central banking system of the United States. It was created on December 23, 1913 with the enactment of the Federal Reserve Act, largely in response to a series of financial panics, particularly a severe panic in 1907...

.
The Federal Reserve Board of Governors

Federal Reserve Board of Governors

The Federal Reserve Board of Governors is the main governing body of the Federal Reserve System. It is charged with overseeing the 12 District Reserve Banks and with helping implement national monetary policy. Governors are appointed by the President of the United States and confirmed by the Senate...

, under Regulation Y (12 C.F.R.