If you want to send email to your EXTERNAL (non ISP email server) then you have to open Mail, go to Mail preferences, look on the pop-up for 'Outgoing Email Server(SMTP)" and use the pull down to select 'Edit Server List'. A window will come down and then highlight your non ISP email server. If the port is set to 25 (for an external email server) change it to port 587 and try that to see if it works.
Also open Mail then go to the menu item "Window" and during the pull down select 'Connection Doctor'. Tell us what it says to help you.