Like the council is called Ahl al-Hall wal-Aqd who are a body of scholars, community leaders, and influential figures who represent the Ummah, I think they have tiers and ranks and stuff but no super sure.
The process of 'election' is consultation which is called Shura, like which is when the council reach a consensus like "okay XYZ is better option to lead the ummah (or this part of the ummah)".
The "vote" (not really) is Bay'ah. Like oath of alleigence. By the end every "vote" must go to the one who is supposed to rule. The ruler or his supporters can urge the remaining people to "vote" for him or given reason of why they aren't "voting" for him.
This whole system is called Siyasah Shar'iyyah which can be translated as just "The Islamic/Sharia Political System".
I may be wrong on Bay'ah part or make mistakes so take it with a grain of salt. I am just a layman who heard a few things.