On June 26 in 1934, President Roosevelt signed the Federal Credit Union Act. The country was still reeling from the banking crises of 1933. Many people's life savings were destroyed when banks failed, and the Federal Credit Union Act's was designed "to make more available to people of small means credit for provident purposes through a national system of cooperative credit, thereby helping to stabilize the credit structure of the united States."
Unlike banks, credit unions are classified as non-profits exempt from taxes. As a cooperative, their customers--called "members"--are also their owners. Monthly statements from credit unions show dividend payments rather than interest.
When the FCUA was first enacted, credit union members were limited to those with a "common bond of association" such as those who shared an occupation (eg., San Francisco Fire Fighters), or those within a well-defined community or rural district (eg. 7th Farm District of Minnesotta)
Starting in 1985, the National Credit Union Administration (a regulatory agency) instituted a policy that allowed credit unions to expand their membership requirements, eliminating one of the primary differences between banks and credit unions.
In 1998, banks complained all the way to the Supreme Court that "diversified" membership violated the Federal Credit Union Act, and won. Their victory was short-lived. Within a year, Congress had written a new law that allowed the expanded membership to continue.
The tension between banks and credit unions continues today. Credit unions argue by serving "people of small means" they fill a critical gap deserving of the tax subsidy. Bankers like David L. Tapp (former President of Overton Bancshares) contend credit unions "should either be allowed to have a common bond or be forced to pay taxes...If credit unions paid taxes and had the same regulatory burden banks do, I'd compete with them every day of the week.''
American Banker Nov 2012 article on credit union lobby requesting an increase to their lending cap
Edgar Callahan, former head of the NCAU dies at 80
April 2013 version of the Federal Credit Union Act
NYT 1997 article: As credit unions boom financial rivals cry
Website arguing against taxing credit unions