There were actually four presidents whose terms in office were technically in the 1920s. President Woodrow Wilson's second term ended in 1921. Wilson was followed by Warren G. Harding, who served from 1921 to 1923, when he died while serving as president. Harding was followed by Calvin Coolidge, who was president from 1923 until 1929, and who was succeeded in the Oval Office by Herbert Hoover, whose tenure ran from 1929 to 1933, when he was replaced by Franklin Roosevelt.
That Harding died while in office is sad; that his Administration ended as soon as did was good. During his time in the White House, corruption reigned supreme, with the Teapot Dome scandal, in which Harding's secretary of the Interior, Albert Fall, leased major government-owned oil fields to a private company with political connections to the Administration. The fact that the oil leases were granted rather than being competitively bid provided the basis for the scandal that subsumed Harding's time in office.
Harding's tenure as president, other than corruption, was actually fairly positive in terms of his positions on civil rights and his economic policies that cut unemployment down considerably. Harding was also way ahead of his time in his advocacy of an International Court -- something that would come into existence until the next millenium.
Calvin Coolidge had been Harding's vice president, but was not tainted by the scandals that rocked his predecessor's Administration. In fact, he is credited with restoring dignity and integrity to the White House. He was a small-government conservative who succeeded in shrinking the federal government before leaving office in 1929.
Herbert Hoover, who succeeded Coolidge, has the ignominy of being President of the United States when, in October 1929, the stock market crashed, precipitating the Great Depression. Needless to say, his one term in office was noteworthy for his efforts at reversing the economic decline that swept the nation -- efforts that, in the case of the Smoot-Hawley tariff bill, are actually credited with prolonging the Depression. His public works projects, however, helped pave the way for Roosevelt's New Deal.