The Constitution gave Congress the power "to coin money and regulate the value thereof." A later clause also gave Congress the power to pass any laws "which shall be necessary and proper for carrying into execution the foregoing powers." Hamilton reasoned that in order to regulate the value of currency, the federal government needed to control the supply of currency. The only way to do that was to establish a national bank.
Democratic-Republicans felt this would give far too much control and power to the federal government and the wealthy elite at the expense of the common workingman. They maintained that the Constitution said nothing about creating a national bank controlled by the federal government. Hamilton replied that the power to do so was implied by the "necessary and proper" clause. This became the Doctrine of Implied Powers. The Federalists eventually got their Bank of the United States after some compromising on this and other issues, such as redeeming government bonds at full face value and federal assumption of war debts, by agreeing to move the site of the new capital city farther south, which the Democratic-Republicans wanted. This is how the nation's capital came to be located on the banks of the Potomac River in what is now Washington, D.C.
The biggest disagreement between the two parties was over Judicial Review. Judicial Review is the power of the courts to declare an act or action unconstitutional. Federalists believed that the federal courts had this right. Democratic-Republicans did not; they felt that the states had that right.