I know this question has been asked ages ago, but I thought I'd drop a 2020 answer here so that it can potentially help new visitors.
Please note:
- This answer serves as a generic answer and will require you to edit some of the details according to form inputs that you are using.
- You also need to update the email address in the headers etc. to the one connected to your domain.
- This solution assumes you're using Google Recaptcha. If not, then just delete the part about "Google recapthca".
- This script has added security and validation that shouldn't be removed.
- If you're going to use Sweet Alert then you should install it into your website/app, either via CDN or NPM.
Some Javascript to create custom Sweet Alert alerts that trigger on mail send:
// Custom SweetAlert alert that gets triggered on email send
function enquirySent() {
    swal({
      title: "Email sent!",
      text: "Thank you for your email. We'll be in contact ASAP.",
      icon: "success",
      button: "Okay",
    });
}
function enquiryNotSent() {
    swal({
      title: "Oops!",
      text: "Your email was NOT sent due to an error.",
      icon: "error",
      button: "Okay",
    });
};
The PHP script to send the mail:
<?php
    if (isset($_POST['submit'])) {
        // For the Google recaptcha
        $curlx = curl_init();
        curl_setopt($curlx, CURLOPT_URL, "https://www.google.com/recaptcha/api/siteverify");
        curl_setopt($curlx, CURLOPT_HEADER, 0);
        curl_setopt($curlx, CURLOPT_RETURNTRANSFER, 1); 
        curl_setopt($curlx, CURLOPT_POST, 1);
        $post_data = [
            'secret' => 'YOUR CAPTCHA SECRET KEY',
            'response' => $_POST['g-recaptcha-response']
        ];
        curl_setopt($curlx, CURLOPT_POSTFIELDS, $post_data);
        $resp = json_decode(curl_exec($curlx));
        curl_close($curlx);
        // Google recaptcha end
        // Form details (sanitized)
        $name = htmlspecialchars($_POST['name']);
        $surname = htmlspecialchars($_POST['surname']);
        $email = htmlspecialchars($_POST['email']);
        $message = htmlspecialchars($_POST['message']);
        // Mail headers and details
        $email_from = 'youremail@yourdomain.com';
        $email_body = "You have received a new message from the user $name $surname.\nHere is the message:\n\n".$message;
        $headers = "From: $email_from \r\n";
        $headers .= "Reply-To: ".$email."\r\n";
        $headers .= "Return-Path: ".$email."\r\n";
        $headers .= "MIME-Version: 1.0\r\n";
        $headers .= "Content-type: text/plain; charset=iso-8859-1\r\n";
        $headers .= "X-Priority: 3\r\n";
        $headers .= "X-Mailer: PHP". phpversion() ."\r\n" ;
        $error = false;
        // Some more input validation/sanitizing
        if (!preg_match("/^[a-zA-Z ]*$/",$first_name) && $first_name!="") {
            $error = true; 
        }
        if (!preg_match("/^[a-zA-Z ]*$/",$last_name) && $last_name!="") {
            $error = true; 
        }
        if (!filter_var($email, FILTER_VALIDATE_EMAIL) && $email!="") {
            $error = true;
        }
        function IsInjected($str) {
            $injections = array('(\n+)',
                   '(\r+)',
                   '(\t+)',
                   '(%0A+)',
                   '(%0D+)',
                   '(%08+)',
                   '(%09+)'
                   );
                       
            $inject = join('|', $injections);
            $inject = "/$inject/i";
            
            if (preg_match($inject,$str)) {
              return true;
            } else {
              return false;
            }
        }
        if (IsInjected($visitor_email)) {
            echo "Bad email value!";
            exit;
        }
        // Sending the email
        if ($error == false) {
            $to = "youremail@yourdomain.com";
            $subject = "Enquiry from website";
            mail($to, $subject, $email_body, $headers);
            // Calling the email sent / not sent alerts
            echo '<script type="text/javascript">',
                'enquirySent()',
                '</script>';
        } else {
            echo '<script type="text/javascript">',
                'enquiryNotSent()',
                '</script>';
        }
    }
?>