You've got three php tags and they're ending in very odd places, try this:
Code:
<?php 
$field_name = $_POST['cf_name']; 
$field_phone = $_POST['cf_phone']; 
$field_email = $_POST['cf_email']; 
$field_message = $_POST['cf_message']; 

$mail_to = 'katie@rrc.la'; 
$subject = 'Message from RA website'; 

$body_message = 'From: '.$field_name."\n"; 
$body_message .= 'E-mail: '.$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); 
?>
<script type="text/javascript">
if (<?php $mail_status ?>) { 
        alert('Thank you for the message. We will contact you shortly.'); 
        window.location = 'contact.html'; 
} 
else {
        alert('Message failed. Please, send an email to katie@rrc.la'); 
        window.location = 'contact.html'; 
} 
</script>