How is a private email domain useful?

One major use is for businesses that want to appear professional by having their email addresses be @companyname.com rather than @gmail.com for example.

Some people use them so all family members can have the same domain; for example bob@lastname.com; sue@lastname.com, etc.

A private domain can help you to have very short email addresses and to use your own name. If you are using a domain with millions of people, for example a mac.com address, you may not be able to use your own name, as someone else will already have claimed this address.