Did Grover Cleveland become president twice? How?

Grover Cleveland did indeed become President of the United States on two separate occasions. His first term of office was from 1885 to 1889; his second was from 1893 to 1897.
Cleveland's first election victory was largely the result of weariness with the Republican Party, which had occupied the presidency since Abraham Lincoln first took office in 1861. Successive Republican Administrations had been mired in corruption, and their candidate in the 1884 election, James G. Blaine, did little to dispel the reputation of the GOP as the party of sleaze.
Cleveland, on the other hand, had an unimpeachable reputation as a redoubtable opponent of corruption, a reputation he'd gained from his short but successful stint as Governor of New York. This was clearly a big factor in his election victory. Blaine also made the big mistake during the campaign of alienating Catholic voters, who were becoming increasingly important in the large cities of the East and Midwest.
In 1888, however, it was a different story; Cleveland was narrowly defeated by his Republican opponent, Benjamin Harrison. Throughout the campaign, Cleveland was hampered by his support for free trade, which was generally unpopular with American business and industry. Harrison's support for maintaining the tariff allowed him to pick up the votes of many Northern Democrats, disaffected as they were by Cleveland's stance on the tariff.
Nevertheless, Cleveland managed to stage a remarkable comeback four years later. Elected to the Presidency for the second time, he benefited from a split in his opponents' ranks. As well as the Republicans, Cleveland faced the Populist Party, which took votes from the GOP in Midwestern and Western states such as Kansas and Nevada, respectively. Though a clear winner in the Electoral College, Cleveland actually won fewer votes than his Republican and Populist opponents combined. Nevertheless, it was a remarkable victory, the last that the Democrats would enjoy at the presidential level for another twenty years.