The principle objective of the Resumption Act of 1875 was to provide for the resumption of specie payments on greenbacks, the fiat paper money born of the Civil War, which was still current in the 1870s.
The act had three important sections. The first section provided for the retirement of the fractional paper currency that been current since the Civil War. The fractional paper currency was in denominations of 10, 25, and 50 cents. The act of 1875 provided for the issuance of subsidiary silver coin to replace the fractional  currency. This provision was a bow to the silver interests because only gold coins circulated at the time. The second section got rid of seigniorage, or mint charges, on the coinage of gold, a provision that pleased the mining interests. The third section of the act removed limitations on the total number of banknotes that national banks could issue, a provision that met the demand for what then was called free banking. The treasury was to retire
greenbacks in an amount equal to 80 percent of the increase in national banknotes, until greenbacks in circulation fell to 300 million.

The third section took up the heart of the legislation, the redemption of greenbacks. After January 1, 1879, greenbacks were redeemable in coin when brought to the assistant treasurer at New York in sums no less than $50. The act also authorized the secretary of treasury to “issue, sell and dispose of, at not less than par, in coin” any of the bonds authorized under existing legislation.
A certain amount of pessimism surrounded the Resumption Act of 1875. Many opponents felt that resumption was not feasible, that people would show up in mass to exchange greenbacks for gold, that it would trigger an unbearable contraction of the money supply, and that Congress would not stand firmly in favor of resumption. In 1878 a bill to repeal the Resumption Act failed to pass Congress by a narrow margin, and Congress did raise from 300 million to 346 million the maximum number of greenbacks that could remain in circulation. Nevertheless, the expected eagerness to exchange greenbacks for gold had been overstated, and resumption took place without difficulty.
The term “coin” in the legislation was generally assumed to refer to gold coins. In 1878, the Bland–Allison Silver Purchase Act made silver legal tender, opening up the possibility that bonds sold to raise coin—gold coin—could be redeemed in silver. Advocates of silver felt that redemption of bonds was legitimate, but the proposal aroused strong opposition. President Hayes in his veto message on the Bland–Allison Act (the act was passed over a presidential veto) cited the large number of bonds the government had sold. He noted that the bonds were sold for gold, and that the bondholders expected the bonds to be redeemed in gold, and would not have bought the bonds otherwise. In the words of his veto message:

National promises should be kept with unflinching fidelity. There is no power to compel a nation to repay its debts. Its credit depends on its honor. The nation owes what it has led or allowed its creditors to expect. (Watson, 1970, 154)
Despite the provisions of the Bland–Allison Act making silver legal tender, the U.S. government maintained its commitment to redeem public bonds in gold.
The Resumption Act of 1875 is one of the important pieces of coinage legislation in U.S. history because it ended an era of fiat money.
See also: Bland–Allison Silver Repurchase Act of 1878, Free Silver MovementGreenbacks
Carothers, Neil. 1930/1967. Fractional Money.
Hepburn, A. Barton. 1924/1967. A History of Currency in the United States.
Meyers, Margaret G. 1970. A Financial History of the United States.
Ritter, Gretchen. 1997. Gold Bugs and Greenbacks: The Antimonopoly Tradition and the Politics of Finance in America.
Watson, David K. 1970. History of American Coinage.