The summoner makes a journey and stops at numerous temples with their guardians. At these temples, the summoner prays to the fayth of the temple and receives an aeon. After being granted assistance by the aeons acquired from the temples, the summoner makes their way to the temple in the Zanarkand Ruins.
Upon completing the trials of this temple, they are required to choose one of their guardians to become the Final Aeon, who is used in the battle against Sin, a battle in which the summoner ultimately dies, regardless of the battle's outcome. The time period after this event is called the Calm, and is a time where Sin is no more. After this Calm, however, the Final Aeon becomes the new Sin and starts the cycle anew. Each Calm only lasts for about 10 years.
Yuna became the first summoner to break the final process in the pilgrimage. Yuna did not receive her Final Aeon, as she was not willing to sacrifice one of her loved ones. She instead traveled inside Sin in order to defeat the source, Yu Yevon, the creator of Sin, thus ending the need for the pilgrimage and bringing an Eternal Calm, which could not have been achieved had she received the Final Aeon.
A pilgrimage is a journey or search of moral or spiritual significance. Typically, it is a journey to a shrine or other location of importance to a person's beliefs and faith.