It's never been done since, because the 22nd amendment, passed in 1947, limits presidents to two terms.
Before FDR, it was sort of "the done thing". Washington declined to serve a third term and Jefferson wrote that "if some termination to the services of the chief Magistrate be not fixed by the Constitution, or supplied by practice, his office, nominally four years, will in fact become for life."
Other presidents have attempted to gain a third term - the Wikipedia article on the 22nd amendment cites Cleveland, Grant and Wilson as examples, but for various reasons it didn't happen.
What made him special was the era he was elected in during the Great Depression. WW II was a factor but he won three elections before the US entered the war (32,36,and 40). When FDR came into the scene after two years of the GD devastating lives and with no help in sight, people looked to the government and a leader that would help relieve the situation. People traveled all over the country looking for work. Many stood in bread lines for hours so they could eat that day. The Hoover administration had some relief programs but it was pretty much a spit in the bucket. FDR promised change and had a successful recorded with progressive programs as gov. of NY. Once elected, New Deal legislation began to roll out which included things that we may take for granted today; unemployment relief, collective bargaining rights,social security, rural electricity, GI bill, and let's not forget the eventual passing of the 21st adm., nullifying prohibition. Also during his first three terms he enacted massive public works projects and banking reforms that people related to and were convicted there was someone representing their views. There were even citizens and politicians that felt he wasn't going far enough. (See Huey Long and Father Coughlin.) But yes he was special in some regards because he was the hope for brighter future for America in a working class. The New Deal did not fix the GD but brought much needed relief to millions of people. You can call this buying votes or being proactive to the needs of people instead of the business elite. The people (most) loved him for it.