AJAX is not a language, but a technique.
Which scripting language is very popular now?
Client-side: javascript is most popular, since all modern browsers support it (unless user has turned it off) and it doesn't need a plugin to run; flash - most of the users has it installed
Server-side: PHP, ASP, JAVA, ... (order doesn't represent popularity)
How to identify whether a Scripting language is Server side Scripting Language or Client Side Scripting Language?
If it is executing in your browser - client-side.
Javascript, JAVA, Flash can execute on client side and on server side.
If it is executing in your browser - client-side.
Instead of "your browser" I would say "your computer", that's more accurate because of widgets, gadgets, AIR, etc.
Client side scripting language (for websites) should be used to improve user experience, good example would be
lightbox.
Server side scripting language is needed when you want dynamic content (probably this isn't accurate, but couldn't think of anything better).