ReCaptcha is Google captcha product.
To use it you need generate public and private keys for your site here.
Public key you set in key of field.
Private key use on server when send to google POST request with needed params.
On server except of all form params it is send g-recaptcha-response
You need to send it to google url too, as response property.
More about it you can read after generate your ReCaptcha on it's page.
Some more about returned params from google here.
When you will try sample, remember that it needs that your server support ssl.
On local server it usually is not enabled.



This sample is on php and shows all simplicity in detail.
For ReCaptcha there are many codes on github, on php too with some better samples.

//login-recaptcha.php

$response = $_POST['g-recaptcha-response'];

$url = 'https://www.google.com/recaptcha/api/siteverify';
$data = array('secret' => 'YOUR PRIVATE KEY', 'response' => $response);

// use key 'http' even if you send the request to https://...
$options = array(
    'http' => array(
        'header'  => "Content-type: application/x-www-form-urlencoded\r\n",
        'method'  => 'POST',
        'content' => http_build_query($data),
    )
);

$context  = stream_context_create($options);
$result = file_get_contents($url, false, $context);

echo $result;