If your mailbox is full, you can login to your Webmail account and delete the emails off of the server.
Deleting messages off of the server is a two step process - first you delete the message, then you purge the deleted messages (this is similar to deleting messages off of your PC/Mac - you delete the file by sending it to the Trash/Recycle Bin, then you empty the Trash/Recycle Bin).
To do this, go to:
Username: email@example.com (this is your full email address)
Password: **this is whatever you set it as, if you do not know/remember your password, you can login to your cPanel account and reset the password (under Email Accounts).
Click on one of the mail reading icons (using Horde for this example) and click Login. Click the + sign next to Mail (from the list on the left hand side) and click Inbox. There you will see a list of messages (you may have it set to "Hide Deleted" messages in which case you can show them by clicking on "Show Deleted" on the upper left hand side of the page. Place a checkmark in the box next to the messages you wish to delete and click the Delete link.
When finished, click on the "Purge Deleted" link on the upper left hand side of the page.
Here is a video tutorial that might help assist you further: http://www.cpanel.net/media/tutorials/emailaccounts.htm