Private quota is that number of seats that are reserved for candidates those who are domiciled in the state where the private mbbs collage is located . The fees for this seats is slightly on the lower side and is affordable by the middle class families mostly. But never the less it's costly.And management quota seats are those seats which are reserved for the college itself that means they can admit anyone who can meet their given guidelines ( mostly it's the moneytary criteria that you have to fulfill but it's different in the case of colleges like St John's and CMC Vellore).Now fees for management quota seats is high and it can only be afforded by the rich people.
Hope this helps
Amartya Chatterjee, MBBS from Malda Medical College and Hospital (2022)
Source: Quora Digest