A city council is a town council 
And what is a town council?
They are like local government
And are they elected or designated by someone?
However it is generally only those local government areas which have been specifically granted city status
And how are city status granted?
City council
A city council, town council, town board, or board of aldermen is the legislative body that governs a city, town, municipality, or local government area.
Australia.
Because of the differences in legislation between the states, the exact definition of a city council varies. However, it is generally only those local government areas which have been specifically granted city status (usually on a basis of population) that are entitled to refer to themselves as cities. The official title is "Corporation of
councils of principal areas that have been granted city status.
- One of the three community councils, with limited functions, that have been granted city status.
United Kingdom Scotland.
A city council is the council of one of four council areas designated a City by the Local Government etc. (Scotland) Act 1994.
The three cities which are not council areas have no city council.
United Kingdom Northern Ireland.
Belfast City Council is now the only city council. Since the local government reforms of 2015