Sending Email from SharePoint
You can configure a task list, to send an email if you assign a task to somebody. This works fine if this somebody has an internal email address.
If you happen to have an account inside the environment of the SharePoint Server, and configure it to have an external email address, sending an email to this account might fail. In the SharePoint Logs there is an entry:
03/12/2007 14:40:42.29 OWSTIMER.EXE (0x059C) 0x14E8 Windows SharePoint Services E-Mail 8gsf High #160009: Die E-Mail-Adresse ‘somebody@somewhere.tld’ ist unbekannt. (something like: the email address ‘somebody@somewhere.tld’ is unknown)
(if not, adjust your logging for email events)
To be able to send emails to recipients outside your environment, you have to grant your SharePoint Server the right to send emails in the virtual SMTP Server of your Exchange Server.