A Democrat was a member or supporter of the Democratic Party, a political party in the United States of America during the 20th century.
The President of the United States in 1947, Harry S. Truman, was a Democrat. After Truman refused to deal with Quark until he learned more about him, Lieutenant General Denning said "that little piano-playing Democrat's not as dumb as he looks." (DS9: "Little Green Men")
Although not specifically referenced on Star Trek as members of the Democratic Party, other Democrats of note include Andrew Jackson, Woodrow Wilson, Franklin D. Roosevelt, John F. Kennedy, Lyndon B. Johnson, Jimmy Carter, William J. Clinton, Hillary Rodham Clinton, Tip O'Neill, Sam Rayburn, and John Glenn.
Denning's reference to Truman being a Democrat is the only time in all of Star Trek that a democratically elected figure is identified by the political party with which he or she is affiliated.