The ajax is going to be a javascript snippet that passes information to a small php file that does what you want. So in your page, instead of all that php, you want a little javascript, preferable jquery:
function fun()
{
$.get('\addEmail.php', {email : $(this).val()}, function(data) {
//here you would write the "you ve been successfully subscribed" div
});
}
also you input would have to be:
<input type="button" value="subscribe" class="submit" onclick="fun();" />
last the file addEmail.php should look something like:
mysql_connect("localhost","root","");
mysql_select_db("eciticket_db");
error_reporting(E_ALL && ~E_NOTICE);
$email=mysql_real_escape_string($_GET['email']);
$sql="INSERT INTO newsletter_email(email) VALUES ('$email')";
$result=mysql_query($sql);
if($result){
echo "You have been successfully subscribed.";
}
if(!$sql)
die(mysql_error());
mysql_close();
Also sergey is right, you should use mysqli. That's not everything, but enough to get you started.