Adding a captcha to a form is very easy with Google ReCAPTCHA. It only takes a few lines of code after you register your domain on the ReCAPTCHA site.

The problem that I discovered is that the captcha appears in Joomla at the very top of the page when it is used on a form. Its being positioned there with CSS in the joomla.css in the templates/css folder. You can modify the code to position it use relative instead of absolute references, or better still just comment out the line and position the captcha in your HTML page.
Search for recaptcha_widget_div in joomla.css and you’ll find something similar to the following code:

/* Compatibility */
#recaptcha_widget_div {position: absolute;top: -140px;}

Comment out the recapthca_widget_div line and the captcha will appear where you define it.