Ajax means "Asynchronous JavaScript + XML". It's not a library but more an architecture, a new approach for web applications based on existing building blocks (XML, CSS, Javascript). The goal of Ajax is to have a more rich user experience in web applications. Ajax is described in an essay from James Garret from Adaptive Path. Google Suggest and Google Maps are good examples of what can be done using Ajax.

On this French article on Ajax from Framasoft, we learn that OpenRico is an open-source JavaScript library for creating rich internet applications and that implement Ajax.

You should also go to Michael Mahemoff's web site on Ajax called Ajax Patterns. It's a tremendous collection of design patterns based on Ajax principles.

Even if Ajax brings probably nothing really new in term of technologies, it has the creditable goal to put user experience in the top prio of web design.