If your message gets sent to the outgoing server (Outlook shows it in the Sent
Items folder), you do not get a non-delivery report after some period of time
(should be at most three days), and your receipient never gets the mesage,
then there is a routing issue somewhere between your mail server and that of
your recipient or you have interfering software on your PC.
The latter is much easier to check. The first thing is to make sure you do
not have an antivirus scanner cauing the problem. Uninstall your AV program
and reinstall it without any mail scanning feature. Doing so does not
decrease your safety. After doing this, try sending another message to see if
it gets through. Do not conclude it did not work until you have waited, say, 24
hours. E-mail is not and never has been instantaneous.
If the message still fails to show up, then you will have to work with your mail
service provider to see if they can find the downstream service interruption.