Email is not send after submit on form

I have problem with sending email from html form. Somehow when i click submit nothing happens. The validation of fields works nevertheless as mentioned before nothing happens when doing submit button. What could be wrong here?

html part:

<section id="contact">
        <div class="container">
            <h3>Formularz kontaktowy</h3>
            <div class="col-md-6 col-md-offset-3 wow fadeInUp" data-wow-delay=".3s">
                <form role="form" id="contactForm">
                    <div class="row">
                        <div class="form-group">
                            <input type="text" class="form-control" name="name" id="name"
                                placeholder="Wpisz swoje imię, nazwisko" required="required">
                        </div>
                        <div class="form-group">
                            <input type="email" class="form-control" name="email" id="email" placeholder="Enter email"
                                required>
                        </div>
                        <div class="form-group">
                            <textarea id="message" name="message" class="form-control" rows="5"
                                placeholder="Enter your message" required></textarea>
                        </div>
                        <button type="submit" id="form-submit" class="btn-block">Wyślij wiadomość</button>
                        <div class="alert alert-success h4 hidden" id="msgSubmit">
                            <strong>Udało się!</strong> Twoja wiadomość została wysłana poprawnie.
                        </div>
                        <div class="alert alert-danger h4 hidden" id="msgError">
                            <strong>Błąd</strong> Prosimy odświeżyć stronę i spróbować ponownie.
                        </div>
                    </div>
                </form>
            </div>
        </div>
    </section>

Scripts:

<script>
        $(document).ready(function () {
            $("#contactForm").on("submit", function (event) {
                if (event.isDefaultPrevented()) {
                } else {
                    event.preventDefault();
                    submitForm();
                }
            });
        });
        function submitForm() {
            $.ajax({
                type: "POST",
                url: "mail.php",
                data: $("#contactForm").serialize(),
                success: function (text) {
                    if (text == "success") {
                        $("#msgSubmit").removeClass("hidden");
                        setTimeout(function () {
                            $("#msgSubmit").addClass("hidden");
                        }, 6000);
                    }
                },
                error: function () {
                    /* You probably want to add an error message as well */
                    $("#msgError").removeClass("hidden");
                }
            });
        };
    </script>

mail.php file:

<?php
//Required field names array
$required = array('name', 'email', 'message');

//Loop over field names, make sure each one exists and is not empty
$error = false;
foreach($required as $field) {
  if (empty($_POST[$field])) {
    $error = true;
  }
}

if ($error) 
    {
       echo "invalid";  //Was empty values should not be reached as html form inspecting it as well
    } 
else 
    {   
        $name = $_POST["name"];
        $email = $_POST["email"];
        $message = $_POST["message"];
 
        $EmailTo = "[email protected]";
        $Subject = $name;
 
        $message .= "nn" . 'From: '. $email;
        $success = mail($EmailTo, $Subject, $message);
 
        //Results catched by AJAX
        if ($success){
        echo "success";
        }else
        {
        echo "invalid";
        }    
    }
?>