"Nursing home" is correct. "Old people's home" is commonly used, but can be seen as insensitive to the residents!

FYI, a convalescent home is not for elderly care; it's for people recovering from an illness or medical treatment. Another term used in the US for this type of lodging is an "assisted living" residence or facility.

You can also say "retirement home", which is less insensitive than "old people's home"