This can apply to people with dementia but also to those with other mental illness.
The Council Tax rules say they must be 'severely mentally impaired', - this includes people who have been diagnosed as suffering with severe and enduring mental health issues including Alzheimer's disease, other forms of dementia and strokes.
For pensioners the person concerned would need to be receiving Attendance Allowance and their council would contact their doctor to check the diagnosis.
This is not means tested and
may mean that person is exempt from Council Tax. This will affect the council tax differently depending on the number of eligible adults in the household.
I did this for my mother and did find it took a while as they contacted the doctor, waited for a reply, etc.
It looks as if most if not all councils have a form on line but you need to check your council's web site as each has their own.
As well as applying to pensioners this would apply to a person in the household who is younger than State Pension Age with a diagnosis of an enduring mental health condition and claiming benefits from a list your council will provide.