Trouble sending e-mail through PHP

I have this very basic HTML:

First Name: <input id="first" type="text" name="first"><br>
Email: <input id="email" type="text" name="email"><br>
<button id="formButton">OK</button>

and jQuery

$('#formButton').click(function(){

var email = $('#email').val();
var first = $('#first').val();
var varData ='name=' + first + '&email=' + email;

$.ajax({
type: "POST",
url: "mail_handler.php",
data: varData,
success: function (){
alert("success");
}
});
});

and here's the PHP mail_handler file:

<?php

$name = $.POST['name'];
$email = $.POST['email'];

mail($email, $name, 'Thank you');
?>

When I try to send an e-mail to myself using this, I get the success alert, but receive no e-mail.

I searched the forum and saw that I might have to include this somewhere, but if so, where?:

<?php
require_once "Mail.php";
$from = "Sender <[email protected]>";
$to = "Recipient <[email protected]>";
$subject = "This is a test email sent via php";
$body = "This is a test email";
$host = "mail.HostingAccountDomain.com";
$username = "[email protected]";
$password = "email_password";
$headers = array ('From' => $from,
'To' => $to,
'Subject' => $subject);
$smtp = Mail::factory('smtp',
array ('host' => $host,
'auth' => true,
'username' => $username,
'password' => $password));
$mail = $smtp->send($to, $headers, $body);
?>

Thanks for any and all help in resolving this issue.

 

Update: I included this code in my home.php file, and I filled in the placeholder information with the required info. Now whenever I load my website, I get an e-mail about 10 minutes later from postmaster saying this is a test e-mail, but I do not get any e-mails from filling out my form and clicking the ok button.

I have tried sticking in the require_once php code inside of my mail_handler.php file, so that way it gets executed at the same time as the php code related to my contact form, but now I am receiving no e-mail at all.