That means that you have too much mail stored on the mail server, and when you exceed your quota, your e-mail account will probably start bouncing e-mail. You need to figure this out quickly to avoid missing important e-mail messages.
I'm assuming that you are using Mail for your e-mail. If not, you're going to need to specify how you are accessing your e-mail.
In Mail, choose Preferences from the Mail menu. In the window that opens, click Accounts. You should see a list of the e-mail accounts you have set up, and underneath will be something that identifies the account type. It will say something along the lines of IMAP, POP, Exchange, iCloud IMAP, etc. What does that say?
If it's IMAP, you say you have deleted everything, but have you emptied the trash in Mail? If not, choose Mailbox -> Erase Deleted Items -> In All Accounts.