The Bull Moose Party was the name of the Progressive Party in the United States in 1912. This party splintered off of the Republican Party as it divided itself between progressivism and conservativism, with the progressives leaving and forming their own party.
To understand the reason for this split, one has to look back at the time of 1912. Teddy Roosevelt had been the Republican president from 1901 to 1909, using a fair amount of progressive policy to usher in a time of vast economic boom and of widespread social change that would largely last until the Great Depression of the 1930s. He was wildly popular as president, and thus his handpicked successor William Howard Taft won the nomination and the election easily in 1908, becoming president in 1909.
Taft, however, let down Roosevelt's progressive ideals and instead was a very conservative president. He basically left the machine running as it was and didn't bother to start new social and economic programs to take advantage of the rampant economy and widespread social change. This eventually started to divide the Republicans into two groups: the progressives and the conservatives.
With the election of 1912 looming, the Republicans, with a very healthy economy and the ongoing success of Roosevelt's social programs, seemed unstoppable. But there was dissent inside the party. The progressive Republicans strongly encouraged the progressive Roosevelt to challenge the conservative Taft for the presidential nomination. Rather than doing this, Roosevelt took a different route: he formed his own Progressive Party and gave it the mascot of the bull moose, hence the common name for the party, the Bull Moose Party.
Naturally, the party to benefit the most from this split was the Democratic Party, the other major party of the day. With the split in the Republican front pretty much down the middle, the Democrats, behind Woodrow Wilson, ran to a reasonably-sized victory over second place finisher Teddy Roosevelt in the 1912 presidential election. With this grand failure, Teddy Roosevelt retired from politics, and with him, the Bull Moose Party largely died as well, with many of the converts going back to the Republican Party.
The great lesson of the Bull Moose Party is the same lesson that the (soon to be) Reform Party demonstrated in 1992: when two parties squabble over a slight majority of voters, the very large minority in favor of the third party is going to carry the election. Even with a hugely popular figure like Teddy Roosevelt heading the party, it wasn't enough to secure enough of the split Republican vote to take the election.
The Bull Moose Party has a role in the history of America as perhaps the strongest example of why a two-party system works here: any breakage in one of the parties guarantees success for the other rather than a strong three horse race.