Sometimes forwarding email from your domain to an existing address makes more sense than having a separate email account. For example, you may have a special address for an event which is used infrequently or only for a short period of time.
However, if you forward your email to a Gmail account it may not work. Here are some of the possible causes and solutions.
If you send yourself a test email and it does not arrive
When you set up your forwarder the first thing you probably did is send yourself a test message to make sure it was working. If this did not arrive then it does not mean your forwarder is faulty. Try sending an email to your forwarding address from another email account (not the Gmail one you are forwarding it to). It should arrive in your inbox.
Here is the reason why this happens:
If you forward mail from firstname.lastname@example.org on another host to a Gmail account then any mail sent from that gmail address to email@example.com will be dropped by Gmail without generating an error message. This is to prevent the risk of forwarding loops. Emails sent from a different Gmail address or from any other email address will get through. This is a weird rule of Gmail's and not part of standard email protocols.
Possible solution (unverified):
If you add your forwarded address as a sending address in your Gmail account and verify it there is some evidence that this disables this block. The evidence I have for this is that the forwarders I have to my own Gmail account that are also set up as external sending addresses in Gmail are able to send mail to themselves.
If your forwarder is not working at all
If you have checked the configuration at the hosting company providing the email forwarder and tried sending from a test message from a different address and it still fails then then the most common reason is that the hosts mail servers are blacklisted by Google. Unfortunately, Google do not publish a list of what they are blocking as this is constantly changing, but in my experience the bigger the hosting company the greater the likelihood of them being blocked. For example Enom's mail servers have been regularly blocked by Gmail over the past few years.
I hope this helps someone work through these issues which are known to affect web hosts using Cpanel and other control panels.