The "Unauthorized relay" error means that your ISP's mail server isn't configured to accept mail for *@mysite.com.
Your ISP is configured to handle mail for *@isp.com, and it won't accept email for any other addresses. Even if it did accept the email, it knows your email address as something like email@example.com - how would it know where to send emails to firstname.lastname@example.org?
Email is hosted just like websites are. Your ISP will not automatically host mail for you. Some ISPs will, either for free or for a small charge, but it's something you need to setup with them. If you have already set that up with them, you need to contact your ISP's tech support to give you the correct MX records to use.
I prefer to stay away from having my ISP host my email - it makes it harder to change ISPs even if something better comes along. Instead you should check if your web hosting provider also does email, or sign up for something like Google Apps for your Domain, which is free for up to 10 users. No matter who you get to host your email, they should give you the correct servers to put in the MX records, as well as what POP3 and SMTP servers to use in your mail client.