By default Exchange will only receive emails being sent across the Active Directory domain, i.e. network.mydomain.local. Most Exchange Servers are set up to receive emails for internet domains, i.e. mydomain.com. Basically there are 3 things that need to performed in order to start receiving emails from these other domains:

1. Add the domain to Accepted Domains
2. Change the default Accepted Domain
3. Configure the Default Receiving Connector

1. Add domain to Accepted Domains

In order to receive emails for a specific domain Exchange must first have the domain added to the list of Accepted Domains. To do this perform the following:

1. Open the Exchange Management Console
2. Under Organization Configuration select either the Hub Transport or the Edge Transport
3. Select the Accepted Domains tab
4. Select New Accepted Domain
5. Enter the Name and the accepted Domain
   Name : MyDomain
   Accepted Domain : mydomain.com
6. Select which type of relay you are performing. If the emails are not passing through any other email server then select Authoritative Domain
7. Click on New

2. Change the Default Accepted Domain

By default, the Default Accepted Domain will be that of the Domain that the Exchange is currently running on.
i.e.  network.mydomain.local
And every users’ mailbox will have them send out emails using that domain.
i.e.  user@network.mydomain.local

So to have emails address to user@mydomain.com by default you will need to set mydomain.com as the Default Accepted Domain, which can be done by performing the following:

1. Open Exchange Management Shell
2. To view the current Default Accepted Domain type:
   Get-AcceptedDomain
   
   The domain having True for Default is the Default Accepted Domain
3. To change it to mydomain.com type :
   Set-AcceptedDomain -Identity mydomain.com -MakeDefault:$true

4. Confirm that mydomain.com is the default by typing:
   Get-AcceptedDomain

5. To quit type:
   exit

3. Configure the Default Receiving Connector

By default the Exchange Server will block all incoming mail that isn’t sent by the Exchange Server itself. Therefore, to accept all mail servers to deliver mail to the Exchange Server performing the following:

1. Open Exchange Management Console
2. Under Server Configuration select Hub Transport
3. Under Receive Connectors go to the properties of the Default Connector
4. Go to the Permissions Groups tab and check Anonymous users