Renting from a council or housing association is also called 'social housing', 'public housing', or 'not-for-profit housing'.
Rents in not-for-profit housing tend to be lower than in private renting, although the gap is closing. Tenants usually have more security, with long-term tenancy agreements, rights of succession (to pass on the home to family members when they die), and legal protection.
But there's a big demand for this type of housing in many areas. Generally, the more urgent your need-to-move, the better your chance of a home. Some rehousing rules also give increased priority to those who make a contribution to the community.
Some housing is reserved for people over a certain age, to reduce the problems caused by conflicting lifestyles. In some areas you can get a home quickly if you are an older person, but may have to wait much longer if you are young - especially if you don't have a high rehousing priority.
If you swant to go for a council or housing association home, you can register for rehousing and look for available homes on the Manchester Move website. Manchester Move has lots of information about other types of housing too.