McCulloch was the head of the Baltimore Branch of the Second Bank of the United States. At that time, the very idea of a federal banking system was hugely controversial. Many leading political figures such as Thomas Jefferson believed that it undermined the rural economy—especially in the South—and gave too much power to the Federal government.
The General Assembly of Maryland felt much the same way, and it voted to tax all banks not explicitly chartered by the state authorities. As the Baltimore Branch of the Second Bank of the United States had been chartered by the federal government and not the state legislature, it was subjected to the new tax.
McCulloch refused to pay the tax, and he eventually took the State of Maryland to court. When the case came before the US Supreme Court, a majority ruled in McCulloch's favor, holding that Congress did indeed have the power to create the Second Bank, even though there was no explicit authorization for this to be found anywhere in the Constitution. The Court creatively interpreted the "Necessary and Proper" clause of the Constitution in such a way as to give the Federal government certain implied powers, one of which was to establish a federal banking system.
This landmark case was hugely important, not least because it increased the power of the Federal government at the expense of state power, thus constituting a major setback for the radical republican principle of state sovereignty.