How do I delete a full mailbox so I can send emails?

Go into the Hub, hold the finger on a mail message and wait until the right margin opens. On this margin you find a trash can on the bottom. This icon means "delete". If you tap it it will ask if you want to delete the message only from the hub or also from the mail server. You will probably have to delete it also from the server.

If you can't send anymore it can also be related to other full folders, not only to the Inbox. Another candidate is the Sent folder.

Folders get full especially if you like to receive mails with big attachments like images and PowerPoint crap. Delete these mails first and tell your mail partners to avoid sending big attachments.