There isn't a particular name for someone between in 50-60 but someone over 60 is usually called an OAP (Old Aged Person).
A person age 60-69 is a sexagenarian, taken from the Latin for "six."
centenarian
The older a person gets the more they find themselves discriminated against. This is called ageism and it affects millions.
Jeanne Calment is the oldest person in the world, and the oldest female in the world. She is a super-centenarian (a person that is 110 years old or more) who was born in February 21, 1875 and died in August 4, 1997 (aged 122 years, 164 days old).
An 80 year old person can sometimes be called elderly. Or they can be called aged, or even a senior or senior citizen.
tricenarian
Teen.
a person in there prime years. That & or Middle-aged
A Person Between The Ages Of Eighty And Ninety Is Called an Octogenarian. (:
A person Aged 90 to 99 years called a Nonagenarian.
A person between the ages of 40 and 49 is a quadragenarian.
70 - 79 - septuagenarian80 - 89 - octogenarian100 years old - centenarian
A person age 60-69 is a sexagenarian, taken from the Latin for "six."
old
centenarian
A Nonagenarian
Gax