With AJAX the browser page is not reloading, but the data is just sent to the server for processing. The server saves the data or calculates something and sends back the answer. The AJAX javascript either displays the answer (page) or does some action depending on the answer.
I’m starting this post with my favorite. Ajax Contact form 0.9 is old but still now the best with some modifications. What you need to add is a captcha to stop some spam. Version 1.0 you can find in updated post where in empty field we have a shake effect.
Download Ajax contact form
Aajax Captcha
In simply steps you can add an image captcha to you contact form.
Download ajax captcha