Final answer:
Theodore Roosevelt started the Progressive Party, also known as the Bull Moose Party, when he ran against William H. Taft in the 1912 presidential election.
Step-by-step explanation:
The 3rd party that Theodore Roosevelt started in order to run against William H. Taft was the Progressive Party, which is sometimes also referred to as the Bull Moose Party. Roosevelt established this party after his dissatisfaction with Taft's presidential policies and because he was snubbed for the Republican nomination in favor of Taft.
The Progressive Party was a significant force in the election of 1912 and aimed to address various social issues, advocate for more responsiveness of the government to the will of the people, and fight political corruption.
However, its existence was short-lived as it largely disappeared after 1916, with most members returning to the Republican Party. The party did enjoy a brief resurgence in 1924 during Robert La Follette's presidential campaign.