The War of 1812, while no property changed hands, was very important for America. It created a new sense of nationalism in the country--America had faced the world's main superpower and managed not to be defeated. There was an increase in American symbols such as the bald eagle, and America gained a new patriotic song, "The Star-Spangled Banner," though it would not be the national anthem until 1931. Conversely, the War of 1812 saw the demise of the Federalist Party; this party, mainly made of Northeasterners, wanted that region of the country to secede over the war's unpopularity. When news of this agenda leaked, those who supported the "Hartford Convention" were branded as un-American and could not be elected. The Democratic-Republicans were now the main party in America, and under James Monroe, the president who followed Madison, the nation enjoyed the Era of Good Feelings. Also, during this era Henry Clay, one of the War Hawks of 1812, instituted the American System of a national bank, internal improvements, and heavy tariffs on foreign products.
Internationally, the War of 1812 led to the British abandoning their forts in the Old Northwest Territory and more Americans moving into the region postwar. Native tribes in the area, especially the Shawnee, lost their main benefactor in Britain and had to take whatever treaty the United States was willing to give them. The territory rapidly became populous states within a generation of the war's end. The British stopped seizing American sailors at sea and trade increased with the new nation. This led to an increase in American commerce immediately after the war. Also, after the disastrous Canadian campaign at the onset of the war, Americans quit clamoring to occupy Canada and focused on moving West instead of North. The national government also increased the size of the army and navy, which had suffered huge budget cuts under Jefferson and Madison--the War of 1812 made the federal government believe that a strong military was essential to American survival.
Long-term, America gained a new generation of statesmen after this war. Andrew Jackson made his name at the Battle of New Orleans and would become an international figure. His campaign nickname of "Old Hickory" also dates from this period as he was said to sleep on the ground alongside his enlisted men. Not to be outdone, the Whig Party in 1840 brought out William Henry Harrison, hero of the War of 1812's battle of Tippecanoe, in which Tecumseh was defeated. Harrison had common-man appeal because he was said to have been born in a log cabin, but he was really the governor and most wealthy farmer in Ohio. Harrison's presidency is nothing of note, as it only lasted one month and led to his Vice-President Tyler completing the term, but it's hard to imagine him becoming famous if not for the War of 1812.