AJAX is a great unknown for many developers, but for me its use is practically mandatory. It is highly recommended to use it but always in moderation, since its use in excess can damage the performance of our page.
Here you have a compilation of 16 forms in AJAX, You can be inspired by them or even download them and put them on your website by modifying the code a bit. And I seriously recommend that you take a look at it because it is the future.
You have them in Web resources depot, site in which this compilation has been curated.