For your e-mail forwarding to work, you need to add the e-mail used on your web hosting account as a contact in your forwarded email account. For example, Gmail, Hotmail, Yahoo, and most other free e-mails require you to add the email being forwarded from as a contact in their system for it to arrive.
This is a way for the system to know you want this e-mail.
Each service is different for how to add your website email as a contact. It should be easy and self-explanatory for the most part but you can contact them for help as well.
If you are not sure how to add the forward in your CPanel we have a video here: Email Forwarder