If all you want is the code, you can find it here. The server component requires PHP to determine whether or not to process the form.
Here's the quick version of reCaptcha versions.
You've recall the series of distorted letters and numbers you've been forced to solve in order to submit a form? That's the original reCaptcha.
The service has evolved with each iteration making it just slightly easier to prove you're a human. Another early version has you click certain tiles on an image collage, identifying them in one way or another. For instance, this example has you identify tiles with street signs.
A later version was much easier in that all you need to do is click a box. The algorithm accounts for imperfections in human response and mouse control. A bot has far more precision and speed than us mere mortals, and that is what is measured to confirm your humanity.
The latest iteration, version 3, is far more clever in that it doesn't require any user intervention. All you see on the page is a version of the icon below and that's about it. Very convenient.
This page contains all the code and a walk through on how to use it for a PHP-based web form.
Questions, comments, or need some implementation help? Drop me a line.