What might have happened if America had won the War of 1812?

The War of 1812 (1812–1815) was a draw. Your question is interesting but answering it involves more speculation than facts. These types of questions are problematic for historians.
American conquest of Canada would have been the most likely and most significant gain had the Americans won the war. In fact, prior to the war, Americans believed conquering Canada would be easy. Canada had only 300,000 people as opposed to America's 8 million. In addition, Americans mistakenly thought that most Canadians wanted to be part of the United States. American attempts to conquer Canada met with abject failure during the war. In 1814, in fact, the British used Canada as a base for an invasion of the U.S.
Florida might also have been taken a few years earlier if America had won. Even though Florida was under Spanish control, the Americans believed the British had too much influence there and used it to destabilize the southeastern part of the United States. As it turned out, Spain ceded Florida to the U.S. in 1821.