Who was the most effective president from 1800-1824?
None of the presidents of this period were without their struggles, even James Monroe, whose presidency encompassed the period known as the "Era of Good Feelings." While Thomas Jefferson had a very difficult second term, I would argue that his first term accomplishments by themselves made him the most effective president from 1800-1824.
Jefferson referred to his election as the "Revolution of 1800," and while this may have been an overstatement, he did succeed in decreasing the size of the federal government, one of his principle priorities before taking office. He also lowered federal spending and taxes, and replaced many Federalist appointees in government with political allies. Jefferson also nearly doubled the size of the United States with the Louisiana Purchase from France and gained victory in an ongoing conflict with Barbary pirates in the Mediterranean. His second term saw Jefferson struggle to maintain American neutrality in the war between Britain and France. His Embargo Act was, and is, largely viewed as a disaster, but he was successful in keeping the United States out of the Napoleonic Wars.