The Exchange Online Control Panel (ECP) displays your Online Company Domains, which are created via the Microsoft Online Portal (MOP). These domains listed in the ECP can be managed in regards to how mail is delivered, which is relevant in a Mail Coexistence scenario; users online and on-premises who use the same email namespace (i.e. firstname.lastname@example.org). In a Mail Coexistence scenario Online and On-Premises users will continue to email/communicate with each other. Exchange Online must be configured properly, so when an Online person sends to an On-Premises person, Exchange will route the mail back to the on-premises Messaging environment.
Note – This scenario is typically seen when Directory Synchronization is in use. Using DirSync provides the synchronization of users, contacts and groups, which are displayed in the Exchange Online Global Address List (GAL), whether they are activated or not. Online users won't know whether these users are online or on-premises, so you will need to configure your Domains via the ECP so on-premises users can receive mail from online users.
Shared Domain Tyoe = (Mail Coexistence) - On-Premises Messaging is Authoritative
If an Online user sends to a user that is not in Exchange Online, the mail will be routed out via the Internet and delivered via MX records – No NDRs are generated in this scenario, assuming the recipient has an on-premises mailbox.
Hosted Domain Type = (Online Only) - Exchange Online is Authoritative
If an Online user sends to a user that is not in Exchange Online, Exchange Online will generate an NDR and not route the message out to the Internet for potential delivery!
Configure your domain for Shared Domain Type
For more information about mail coexistence in Office 365, visit the following Microsoft website