This tutorial will teach you how to make a simple newsletter in PHP/MYSQL with account activation. This will require PHP that is not running in SAFE_MODE or PHP that at least allows the sendmail function(duh, you can't send mail without this enabled).
The answer? Use a form mailer! A form mailer is a system which takes information that someone types into a form on your website, and turns it into an email which you can read.
Although PHP has built in function mail() to send email, it's quite insecure and use nonobvious arguments. Here is the example of easy-to-use and reliable mail function.
Joining mailing list will entitle you
to receive occasional emails informing you of news and
updates to the site and any special offers that may be
of interest to you.