The Spaniards had built mints all over the colonies, and the Spanish Milled Dollar was ubiquitous. In some colonies it was made the unit of account. When independence was declared, the colonies adopted Articles of Confederation which gave Congress the power to issue paper money, called “continentals.” Here is an example of a continental $30 bill. Notice that it “entitles the Bearer to receive Thirty Spanish milled Dollars or the Value thereof in Gold or Silver.”

A $30 bill, issued by the Continental Congress for Thirty Spanish milled Dollars

After independence was achieved, and the Constitution was adopted, the U.S. did not want to rely on Spanish mints to mint its coins. The U.S. wanted its own mints to mint its own coins, including dollars. To that end, Alexander Hamilton, then Secretary of the Treasury, wrote the Coinage Act of 1792 wherein he tells us exactly what a dollar is:

“Dollars or Units — each to be of the value of a Spanish milled dollar as the same is now current, and to contain three hundred and seventy-one grains and four sixteenths parts of a grain of pure, or four hundred and sixteen grains of standard silver.”
 
This definition of a dollar, 371.25 grains of fine silver, has never been changed, and cannot be changed. The Constitution requires that a dollar be a weight of silver.