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 your 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