The previous posts gave some very strong answers.  I would suggest that one of FDR's greatest accomplishments was proving how important the role of President can be under specific conditions.  I think that if one created a list of Presidents who had to face some of the most severe and difficult conditions, FDR's name would be on the short list.  The economic depression which crippled America was one set of conditions, while another was that no foreign nation could be turned to for help.  Add to this the rise of European Fascism and its threat to democracies all over the world and that few Americans felt confident about their world and their place in it, and one realizes the FDR had to face some difficult challenges.  Within all of this, FDR's major accomplishment was to instill a sense of worth within the American individual.  Whether it was through Fireside Chats or the idea that he, someone who came from wealth and privilege, could identify with the "average Joe," FDR was able to use his power as President to make Americans believe in someting other than their own helplessness.  Presidents since FDR have understood the power and need to identify with their citizens in order to underscore their own political viability.

Basically, FDR did two things.  First, he helped reduce the impact of the Great Depression.  Second, he led the US during most of World War II.  This means that he was president during two of the biggest crises in US history.

People often say that FDR ended the Depression with the New Deal.  Most historians do not agree with this.  They say that the coming of WWII ended the Depression.  However, FDR's policies and his personal abilities clearly helped to stabilize the economy after 1933 and prevent it from getting worse.

Before and during WWII, FDR forged an alliance with Great Britain and (less so) with the USSR.  He skillfully managed this alliance during the war and he was in overall charge of the war effort.  his work clearly helped the Allies to win the war.