Why do I not receive any emails?

Question

Last month I started a project for a good grade. I finally got my website how I want it to be, but I have a problem with my PHP I think.

My HTML is:

    <form id="ajax-contact" method="post" action="mailer.php">
        <div class="field">
            <label for="name"></label>
            <input type="text" id="name" name="name" placeholder="Name" required>
        </div>

        <div class="field">
            <label for="email"></label>
            <input type="email" id="email" name="email" placeholder="Email" required>
        </div>

        <div class="field">
            <label for="message"></label>
            <textarea id="message" name="message" placeholder="Nachricht" required></textarea>
        </div>

        <div class="12u$ field">
            <ul class="actions">
                <li><input type="submit" value="Nachricht senden" /></li>
            </ul>
        </div>
    </form>

    <br>
    <div style="text-align: center" id="form-messages"></div>
    <script src="jquery-2.1.0.min.js"></script>
    <script src="app.js"></script>

And I used this for my PHP:

<?php
// My modifications to mailer script from:
// http://blog.teamtreehouse.com/create-ajax-contact-form
// Added input sanitizing to prevent injection

// Only process POST reqeusts.
if ($_SERVER["REQUEST_METHOD"] == "POST") {
    // Get the form fields and remove whitespace.
    $name = strip_tags(trim($_POST["name"]));
            $name = str_replace(array("\r","\n"),array(" "," "),$name);
    $email = filter_var(trim($_POST["email"]), FILTER_SANITIZE_EMAIL);
    $message = trim($_POST["message"]);

    // Check that data was sent to the mailer.
    if ( empty($name) OR empty($message) OR !filter_var($email, FILTER_VALIDATE_EMAIL)) {
        // Set a 400 (bad request) response code and exit.
        http_response_code(400);
        echo "Oops! There was a problem with your submission. Please complete the form and try again.";
        exit;
    }

    // Set the recipient email address.
    // FIXME: Update this to your desired email address.
    $recipient = "adress@mail.com";

    // Set the email subject.
    $subject = "New contact from $name";

    // Build the email content.
    $email_content = "Name: $name\n";
    $email_content .= "Email: $email\n\n";
    $email_content .= "Message:\n$message\n";

    // Build the email headers.
    $email_headers = "From: $name <$email>";

    // Send the email.
    if (mail($recipient, $subject, $email_content, $email_headers)) {
        // Set a 200 (okay) response code.
        http_response_code(200);
        echo "Thank You! Your message has been sent.";
    } else {
        // Set a 500 (internal server error) response code.
        http_response_code(500);
        echo "Oops! Something went wrong and we couldn't send your message.";
    }

} else {
    // Not a POST request, set a 403 (forbidden) response code.
    http_response_code(403);
    echo "There was a problem with your submission, please try again.";
}
?>

Now I also have liked two Javascript files to make it complete:

My first JS is (app.js) :

$(function() {

// Get the form.
var form = $('#ajax-contact');

// Get the messages div.
var formMessages = $('#form-messages');

// Set up an event listener for the contact form.
$(form).submit(function(e) {
    // Stop the browser from submitting the form.
    e.preventDefault();

    // Serialize the form data.
    var formData = $(form).serialize();

    // Submit the form using AJAX.
    $.ajax({
        type: 'POST',
        url: $(form).attr('action'),
        data: formData
    })
    .done(function(response) {
        // Make sure that the formMessages div has the 'success' class.
        $(formMessages).removeClass('error');
        $(formMessages).addClass('success');

        // Set the message text.
        $(formMessages).text(response);

        // Clear the form.
        $('#name, #email, #message').val('');
    })
    .fail(function(data) {
        // Make sure that the formMessages div has the 'error' class.
        $(formMessages).removeClass('success');
        $(formMessages).addClass('error');

        // Set the message text.
        if (data.responseText !== '') {
            $(formMessages).text(data.responseText);
        } else {
            $(formMessages).text('Oops! An error occured and your message could not be sent.');
        }
    });

});

});

And the other JS (jQuery JavaScript Library v2.1.0. min) found and downloaded from google.

Now I dont understand why I dont receive any mail in my mailbox. (by replacing adress@mail.com with my mail adress ofcourse) I think it also could be my webhost but I don't know for sure.