Wordpress - How to change the default registration email ? (plugin and/or non-plugin)

The new user email is sent using the wp_new_user_notification() function. This function is pluggable, which means that you can overwrite it:

// Redefine user notification function
if ( !function_exists('wp_new_user_notification') ) {
    function wp_new_user_notification( $user_id, $plaintext_pass = '' ) {
        $user = new WP_User($user_id);

        $user_login = stripslashes($user->user_login);
        $user_email = stripslashes($user->user_email);

        $message  = sprintf(__('New user registration on your blog %s:'), get_option('blogname')) . "\r\n\r\n";
        $message .= sprintf(__('Username: %s'), $user_login) . "\r\n\r\n";
        $message .= sprintf(__('E-mail: %s'), $user_email) . "\r\n";

        @wp_mail(get_option('admin_email'), sprintf(__('[%s] New User Registration'), get_option('blogname')), $message);

        if ( empty($plaintext_pass) )
            return;

        $message  = __('Hi there,') . "\r\n\r\n";
        $message .= sprintf(__("Welcome to %s! Here's how to log in:"), get_option('blogname')) . "\r\n\r\n";
        $message .= wp_login_url() . "\r\n";
        $message .= sprintf(__('Username: %s'), $user_login) . "\r\n";
        $message .= sprintf(__('Password: %s'), $plaintext_pass) . "\r\n\r\n";
        $message .= sprintf(__('If you have any problems, please contact me at %s.'), get_option('admin_email')) . "\r\n\r\n";
        $message .= __('Adios!');

        wp_mail($user_email, sprintf(__('[%s] Your username and password'), get_option('blogname')), $message);

    }
}

Note: Overriding pluggable functions cannot be done in the theme functions.php file. WP's pluggable file is already loaded by that point so the function would be defined by WP (i.e. the default). Your custom version must load before this happens, which means you must load it in a custom plugin file.

For 2018 and onwards users:

Since WordPress 4.9.0 there are new filters you can use for this (no need for a plugin anymore):

• wp_new_user_notification_email - customise email sent to User
• wp_new_user_notification_email_admin - customise email sent to Admin

Usage example on email sent to Admin (you can paste this in your theme's functions.php ):

add_filter( 'wp_new_user_notification_email_admin', 'custom_wp_new_user_notification_email', 10, 3 );

function custom_wp_new_user_notification_email( $wp_new_user_notification_email, $user, $blogname ) {
    $wp_new_user_notification_email['subject'] = sprintf( '[%s] New user %s registered.', $blogname, $user->user_login );
    $wp_new_user_notification_email['message'] = sprintf( "%s ( %s ) has registerd to your blog %s.", $user->user_login, $user->user_email, $blogname );
    return $wp_new_user_notification_email;
}

This will not work on functions.php. You need to put this code inside a plugin.

If you don't now to make a plugin for this just use this link.

Don't forget to take the update code of this function form here.