html php Contact form seems like it works, but isn't actually sending the email?

Question

There is an issue with my contact form, and I have not been able to find solutions that work from similar questions.

I found this code with a Google search, and it seems to work fine.. It shows no errors, and the success message appears. But it is not actually sending to my Email when I check! Any one know what is going on here?

Here is the HTML

    <form name="contactform" method="post" action="form.php">
    <table width="450px">
    <tr>
     <td valign="top">
      <label for="first_name">First Name *</label>
     </td>
     <td valign="top">
      <input  type="text" name="first_name" maxlength="50" size="30">
     </td>
    </tr>
    <tr>
     <td valign="top">
      <label for="last_name">Last Name *</label>
     </td>
     <td valign="top">
      <input  type="text" name="last_name" maxlength="50" size="30">
     </td>
    </tr>
    <tr>
     <td valign="top">
      <label for="email">Email *</label>
     </td>
     <td valign="top">
      <input  type="text" name="email" maxlength="80" size="30">
     </td>
    </tr>
    <tr>
     <td valign="top">
      <label for="telephone">Phone</label>
     </td>
     <td valign="top">
      <input  type="text" name="telephone" maxlength="30" size="30">
     </td>
    </tr>
    <tr>
     <td valign="top">
      <label for="comments">Message *</label>
     </td>
     <td valign="top">
      <textarea  name="comments" maxlength="1000" cols="25" rows="6"></textarea>
     </td>
    </tr>
    <tr>
     <td colspan="2" style="text-align:center">
      <input type="submit" value="Submit">
     </td>
    </tr>
    </table>
    </form>

PHP:

<?php
if(isset($_POST['email'])) {

    // EDIT THE 2 LINES BELOW AS REQUIRED
    $email_to = "vpagliuc_capstone@yahoo.com";
    $email_subject = "Contact Email";

    function died($error) {
        // your error code can go here
        echo "We are very sorry, but there were error(s) found with the form you submitted. ";
        echo "These errors appear below.<br /><br />";
        echo $error."<br /><br />";
        echo "Please go back and fix these errors.<br /><br />";
        die();
    }


    // validation expected data exists
    if(!isset($_POST['first_name']) ||
        !isset($_POST['last_name']) ||
        !isset($_POST['email']) ||
        !isset($_POST['telephone']) ||
        !isset($_POST['comments'])) {
        died('We are sorry, but there appears to be a problem with the form you submitted.');
    }



    $first_name = $_POST['first_name']; // required
    $last_name = $_POST['last_name']; // required
    $email_from = $_POST['email']; // required
    $telephone = $_POST['telephone']; // not required
    $comments = $_POST['comments']; // required

    $error_message = "";
    $email_exp = '/^[A-Za-z0-9._%-]+@[A-Za-z0-9.-]+\.[A-Za-z]{2,4}$/';

  if(!preg_match($email_exp,$email_from)) {
    $error_message .= 'The Email Address you entered does not appear to be valid.<br />';
  }

    $string_exp = "/^[A-Za-z .'-]+$/";

  if(!preg_match($string_exp,$first_name)) {
    $error_message .= 'The First Name you entered does not appear to be valid.<br />';
  }

  if(!preg_match($string_exp,$last_name)) {
    $error_message .= 'The Last Name you entered does not appear to be valid.<br />';
  }

  if(strlen($comments) < 2) {
    $error_message .= 'The Comments you entered do not appear to be valid.<br />';
  }

  if(strlen($error_message) > 0) {
    died($error_message);
  }

    $email_message = "Form details below.\n\n";


    function clean_string($string) {
      $bad = array("content-type","bcc:","to:","cc:","href");
      return str_replace($bad,"",$string);
    }



    $email_message .= "First Name: ".clean_string($first_name)."\n";
    $email_message .= "Last Name: ".clean_string($last_name)."\n";
    $email_message .= "Email: ".clean_string($email_from)."\n";
    $email_message .= "Telephone: ".clean_string($telephone)."\n";
    $email_message .= "Comments: ".clean_string($comments)."\n";

// create email headers
$headers = 'From: '.$email_from."\r\n".
'Reply-To: '.$email_from."\r\n" .
'X-Mailer: PHP/' . phpversion();
@mail($email_to, $email_subject, $email_message, $headers);
?>

<!-- include your own success html here -->

Thank you for contacting us. We will be in touch with you very soon.

<?php

}
?>

Thank you!