jquery Remodal e-mail form - php

Question

I'm using the Remodal plugin to show modals on my website. (https://github.com/VodkaBears/Remodal) This works great for my needs. This is the first time I'm working with modals and e-mail forms. I have a script to submit the form, and it looks like this;
<form name="contact" id="contact" class="contact" method="post">
<label for="name">
<span>Name</span>
<input type="text" name="name" required="true" />
</label>
<label for="email">
<span>Email</span>
<input type="email" name="email" required="true" />
</label>
<label for="phone">
<span>Phone</span>
<input type="text" name="phone" required="true" />
</label>
<label for="message">
<span>Your message</span>
<textarea name="message" required="true"> </textarea>
</label>
<label>
<br>
<span> </span>
<input type="submit" class="send" value="Send"/>
</label>
</form>
And this is my script for submitting the form;
$(document).on('click', '.send', function () {
$.ajax({
type: "POST",
url: "mail.php",
data: $('form.contact').serialize(),
success: function (msg) {
$("#thanks").html(msg)
$("form.contact").modal('hide');
},
error: function () {
alert("Error.");
}
});
return false;
});
Mail.php:
<?php
//if "email" variable is filled out, send email
if (isset($_REQUEST['email'])) {
//Email information
$admin_email = "my#email.com";
$name = $_REQUEST['name'];
$email = $_REQUEST['email'];
$phone = $_REQUEST['phone'];
$message = $_REQUEST['message'];
//send email
mail($admin_email, "New e-mail", $message, "From:" . $email);
//Email response
echo "Thank you!";
}
?>
I found this code somewhere, and I'm trying to implement it to my form. It works! However, I don't get any success message or error messages with this. I'm trying to wrap my head arond it, but can't figure it out. Can someone help?