Horace Greeley
Ulysses S. Grant was president from 1869-1879. Ulysses S. Grant ran for reelection against Horace Greeley. (Grant was in office until 1877.)
Horace Greeley
Horace Greeley was a cranky newspaper editor who carried the liberal republican and democratic banners against Grant. He was the founder and editor of the New York Tribune and a vocal critic of President Ulysses S. Grant's administration. Greeley ran for president in 1872 as the nominee of the Liberal Republican Party and the Democratic Party, but lost to Grant.
His opponent in 1868 was Horatio Seymour from New York. In 1872 Horace Greeley, also from New York, ran against him. (Greeley died after the popular election but before the electoral college voted, so it might have been legally interesting if Greeley had won. )Victoria Woodhull also ran in 1872, but she was too young to be president, did not appear on any ballots and did not get any votes, so some doubt that she was a serious candidate.
There seems to be some confusion in your question. Ulysses S. Grant was the 18th President of the United States and served from 1869 to 1877. There was no presidential election in 1972 involving Grant or anyone running as a liberal Republican candidate. Richard Nixon, a Republican, ran against George McGovern, a Democrat, in the 1972 presidential election.
Horatio Seymour from New York was the Democratic opposition to the Republican candidate ,U. S Grant in 1868. Grant was the winner, 219-80, in electoral votes.
He ran against President John Adams
Horace Greeley is your answer. But he didn't win. There was a split within the Republican Party that resulted in a defection of many Liberal Republicans to opponent Horace Greeley. The other major political party, the Democratic Party, also nominated the candidates of the Liberal Republican ticket that year.
The first case of this was Thomas Jefferson, who ran against John Adams.
President Arthur never ran for President.
President George W. Bush ran against Senator John Kerry in 2004. President Bush won. Jim Carey, an actor, never ran for political office and never ran against any President of the USA.