The national government's power to coin money is its authority to issue money for standard usage throughout the United States. The government also has the related power of regulating the value of the money that it makes. This power comes from Article I, Section 8 of the U.S. Constitution, but the national understanding of the extent of this power has changed throughout history.
The Heritage Foundation explains that the phrase "to coin money" refers to Congress's authority to make money from precious metals like gold and silver. The significance of this clause of the Constitution lies in the fact that historically the power to coin money was identified with political sovereignty. The states were unnerved by the acquiescence of political sovereignty to the national government.
Initially, the issuance of paper money was not considered to be part of the national government's power to "coin money." Wikipedia describes how Article 1, Section 10 of the Constitution forbids the states from making anything other than silver and gold coins "legal tender," but it does not define any such prohibition for the federal government. Nevertheless, paper money was long thought to be unconstitutional. It was not until the presidency of Ulysses S. Grant that the Supreme Court upheld the constitutionality of paper money issued during the Civil War.