Javascript wouldn't be right for this — it works mainly on the client side. PHP would be the right choice. If you search on Google, you'll find several tutorials and sample code for mail-sending scripts. A basic template would be something like this:
(on the HTML page)
[codex]
<form action="sendmail.php" method="post">
<input type="text" name="address" value="Enter your e-mail"></input>
<input type="submit"></input>
</form>
[/codex]
(in sendmail.php)
[codex]<?php
$address = stripslashes($_REQUEST['address']);
mail('youremail@example.com',"Mailing list signup","$address would like to sign up for the mailing list");
echo "Thank you for signing up!";
?>[/codex]
You'll want a little more than that (e.g., an actual page that redirects them back to your site instead of just "Thank you for signing up!"), but that's the basic form of it.