What is the difference between child domain and domain controller?

A child domain is another domain under a parent one in an active directory domain hierarchy.
A child domain under a parent first root domain form a Tree.
All Trees exists within a Forest,a forest is the security boundary.
A domain controller is the physical machine that holds the active directory database, it's responsible of authenticating all users within the domain and applying group policies on the machines.

A child domain is another domain under a parent one in an active directory domain hierarchy. A child domain under a parent first root domain form a Tree. All Trees exists within a Forest,a forest is the security boundary. Where a domain controller is the physical machine that holds the active directory database, it's responsible of authenticating all users within the domain and applying group policies on the machines.

Child domain is a part of your main domain, if yahoo.com is your main domain then your sub domain will be mail.yahoo.com.
Domain Controller is simply a physical machine, on which you install your domain.

What are the objective and problem of child and three domain?