Final answer:
The Republicans won the 1896 presidential and congressional elections with William McKinley's victory over William Jennings Bryan, who was backed by both the Democrats and the Populist Party.
Step-by-step explanation:
The political party that won the 1896 presidential and congressional election was the Republicans. During this election, the Republican candidate William McKinley defeated the Democratic candidate William Jennings Bryan, who was also supported by the Populist Party. The Republicans saw victory not just in the presidency but also made substantial gains in Congress. Despite garnering significant support from the South and Midwest farmers, the Democratic and Populist alliance failed to overcome the Republican stronghold. The Populist Party, while influential and demonstrating early success, ultimately could not sustain momentum beyond this period, especially with the loss in the 1896 election. Overall, the Republican victory signaled a preference among voters for economic stability over the systemic changes advocated by the Populists and their Democratic ally.