what does ran for third party ticket mean?
It means ran for a party that was not the Democratic Party and not the Republican Party.
so how does this apply to Theodore Roosevelt
Read, read, read ...
http://www.google.com/#q=teddy+roosevelt+third+party+election
Watch out for Moose tracks.
"Running for a third party ticket" refers to a candidate who is campaigning for elected office as a candidate representing a political party other than the two major parties in a particular country (such as the Democrats and Republicans in the United States). Third parties are typically smaller and less influential than the major parties, and they may have different policy positions or ideologies.
To understand the concept of running for a third party ticket, it's important to recognize the two major parties in the country you're referring to. In the United States, for instance, the major parties are the Democrats and Republicans. When someone decides to run for office, they can do so as a candidate from one of these parties. However, individuals may also choose to run as candidates from a third party, such as the Green Party, Libertarian Party, or any other smaller party.
By running for a third party ticket, these candidates are often trying to bring attention to alternative policy positions or represent a specific group of voters who may feel underrepresented by the two major parties. While it is possible for third-party candidates to win elections, it can be challenging due to the dominance of the major parties and the electoral systems in place. Nevertheless, they play a role in the political landscape by introducing new ideas and perspectives into political discourse.