It may be that your server doesn't have mail() enabled, or that the mail server info is not correct. Also, something in index.html may not be right.
Also on closer inspection I see that this line of your code will break.
$message = "Hello, my name is $_POST['name']";
CHANGE TO:
$message = "Hello, my name is ".$_POST['name'];
ALSO: you call
mail.php
as your action
but you say in your message:
Mailer.php says...
So are your files named right? is it mail.php OR Mailer.php??
That wasn't your question, however. Your question was if you could pass a parameter input on one page to a different page.
To prove this can be done, change the code of
mail.php
to
<?php
$message = "Hello, my name is ".$_POST['name'];
die($message);
?>