The main effect of the War of 1812 on American domestic politics was that it brought about the end of the Federalist Party. This led to a short time in which there was only one political party in the US.
Before the War of 1812, there were two main parties, the Federalists and the Democratic-Republicans (this group was usually just called the Republicans, but it is not the same party as the Republican Party that we have today). The Federalists had dominated the early US, but they had become much weaker by the time the war began. They were, essentially, a part of New England.
The Federalists were strongly opposed to the War of 1812. This was partly for ideological reasons as the Federalists supported Britain and its constitutional monarchy against France and its revolutionary democracy. It was also partly out of economic interest since New England’s economy was based largely on trade with Great Britain. The Federalists were so opposed to the war that states with Federalist governors refused to send militias to help with the war. They spoke against the war and predicted that it would harm the US. In late 1814 and early 1815, the Federalists met at the Hartford Convention. Some extremists in the party talked of secession and the convention proposed a number of amendments to the Constitution that were meant to weaken the national government. When the war ended and the US did not suffer defeat, the Federalists were discredited. They seemed to many people like traitors because they had failed to back the country during a war. The party never recovered from this and soon disappeared altogether.
In this way, the War of 1812 helped to destroy the Federalist Party, which had been the party of Alexander Hamilton and George Washington. This was the main effect that it had on domestic politics in the US.