Parent Domain, First installation of your AD, you call it first forest and first domain(1Server as an AD). For the example as you can see from the picture contoso.com as a parent domain from Jakarta.contoso.com. Child Domain, You can create the child domain after you have deployed your first domain, and the the child name will followed parent domain, consideration to have a child domain is usually for branch office want to all their network resource on their own or they(branch office) want to have separate database from Head Quarter. For the example as you can see from the picture jakarta.contoso.com as a child domain from contoso.com. Different Domain in same Forest, The same like child domain, installed after first domain deployed in a forest but have different name from the parent(totally different name). For the example as you can see from the picture Woodgrovebank.com is a different domain from contoso.com but it is still within the same forest.
Continued Active Directory Concepts Part 3
Richard Mueller edited Revision 3. Comment: Modified title, added TOC, added tag
Ed Price - MSFT edited Revision 2. Comment: title