The first thing that happens is that the council declares a vacancy - this should be done soon after the vacancy occurs. The vacancy must then be notified by a public notice. A poll can then be claimed by 10 electors within 14 days of the notice being made public. If this happens, then an election must take place within 60 days of the notice of vacancy.
If no poll is claimed, then the council may fill the vacancy by cooption. The council should give public notice that it is seeking candidates for cooption and allow a reasonable period for applications. Once the period has ended, the council may make an appointment. The appointed candidate must receive an absolute majority of the votes of the councillors present and voting (including casting vote if any) and if that is not achieved on the first vote, the candidate with the fewest votes is eliminated and another vote taken. This continues until one candidate achieves an absolute majority.