If you are having problems sending emails from your own computer using an email client then start here:
Check your mailboxes on the server and make sure they are not full or that they are not so close to their quota's that the email your trying to send would overfill them. (this is done via cPanel, your website control panel)
Check over your cPanel hosting account as a whole and make sure that the hosting account itself is not "full" and has reached its entire disk allocation limit as this will also cause your mailboxes to have no room left for storage.
If everything looks to be fine, then login via your webmail (yourwebsite/webmail/) and then try sending emails. If that then works, then you know the problem is with your ISP blocking outgoing mail relay. You will need to contact them and ask to have mail relaying enabled on your account.
This is a common practice with a lot of non-business internet providers to prevent their internet services being abused by spammers.
If your using something like Microsoft Outlook or other email clients on your desktop computer, you should get an error message that tells you outgoing relaying is not allowed, or something similar.
You could try changing your outgoing mail port within the email client settings to port 26 instead of 25, but it is possible they may have this blocked too.
Most ISP's will however, unblock this for you if you call them and tell them what your doing and ask them nicely :-)
If you are getting any other error at all, or find your working email suddenly is not working, and you have checked with your Internet Provider that they have NOT just disabled mail relaying, then do let us know.