Refer to the included comments.

contact.php

<?php
/**
 * "Undefined index" means that you're trying to fetch data from a variable
 * that doesn't exist (or isn't "defined"). Looking at your code here,
 * personally, what I would do here, firstly, is check whether your $_REQUEST
 * variables are empty, and if they're not, *then* set your $action variable,
 * rather than before.
 */
if (!empty($_REQUEST['action'])) {
    $action = $_REQUEST['action'];
    ?>
    <form action="contact.php" method="POST" enctype="multipart/form-data">
    <input type="hidden" name="action" value="submit">
    Your name:<br>
    <input name="name" type="text" value="" size="30"/><br>
    Your email:<br>
    <input name="email" type="text" value="" size="30"/><br>
    Your message:<br>
    <textarea name="message" rows="7" cols="30"></textarea><br>
    <input type="submit" name="submit" value="Send email"/>
    </form>
    <?php
} else {
    /** Same principle as with the "action" variable, we'll do the same here. */
    if (empty($_REQUEST['name']) || empty($_REQUEST['email']) || empty($_REQUEST['message'])) {
        echo 'All fields are required, please fill <a href=\'\'>the form</a> again.';
    } else {
        $name = $_REQUEST['name'];
        $email = $_REQUEST['email'];
        $message = $_REQUEST['message'];
        $from = 'From:' . $name . $email . "\r\nReturn-path: " . $email;
        $subject = 'Message sent using your contact form';
        mail('youremail@yoursite.com', $subject, $message, $from);
        echo 'Email sent!';
    }
}