submitting contact form, I don't receive email

I'm adding simple contact form to my website, but I don't receive emails when I submit it.

my html is:

<form action="contact.php" method="post" class="form-container">  
        <label for="name" class="form-title">Name</label> 
        <input id="name" class="form-field" type="text" name="cf_name" required><br/>
         <label for="email" class="form-title">Email</label>
        <input id="email" class="form-field" type="email" name="cf_email" required pattern="[a-z0-9._%+-]+@[a-z0-9.-]+\.[a-z]{2,3}$"><br/>
        <label for="message" class="form-title">Message</label>
        <textarea id="message" class="form-field" name="cf_message" required></textarea><br/> 
          <div class="submit-container"> 
        <input class="submit-button" type="submit" value="Send">
        </div>
    </form>

my php is:

<?php   
 $field_name = $_POST['cf_name'];
 $field_email = $_POST['cf_email'];
 $field_message = $_POST['cf_message'];    

 $mail_to = 'info@littlerivermiami.com';
 $subject = 'Message From: '.$field_name;


 $body_message = 'E-mail: '.$field_email."\n";
 $body_message .= 'Message: '.$field_message;


 $headers = 'From: '.$field_email."\r\n";
 $headers .= 'Reply-To: '.$field_email."\r\n";

 $mail_status = mail($mail_to, $subject, $body_message, $headers);

 if ($mail_status) { ?>
    <script language="javascript" type="text/javascript">
        alert('Thank you for the message. We will contact you shortly.');
        window.location = 'contact.html';
    </script>
 <?php
 }
 else { ?>
    <script language="javascript" type="text/javascript">
        alert('Message failed. Please, send an email to info@littlerivermiami.com');
        window.location = 'contact.html';
    </script>
 <?php
 }
 ?>

I do see alert message that means the code works, but I don't receive email with filled in info from form.

any ideas why is it happening?