Machina (machines) is what brought Sin at least 1000 years ago. Machina, therefore should never be used. Sin is the price we pay for using machina.
If the people of Spira believe and follow Yevon's teachings, Spira we then atone for the use of machina long ago and Spira will atain purity. It is then when Spira will live their lives without Sin.
The teachings state that a summoner will go to Zanarkand and pray to the fayth and will call forth the Final Aeon and the Final Aeon will take the summoner's life and in the process, destroy Sin, creating the Calm.
More on this section later. While I'm replaying the game, I'll take notes of the beliefs of Yevon and I'll post up more here later. ;)
