I know nothing about flex and air, but Flash, Java FX and Silverlight are all web technologies that essentially do the same thing vying for market share because none of these companies (Adobe, Sun, and Microsoft, respectively) wants to give the other an edge and/or not control the major content delivery platform on the web. That's it in a nutshell. Market speak would probably include something like "rich internet applications" or something like that.
ActiveX was, unless I am mistake, a huge festering security hole, that is largely abandoned even by Microsoft and Java applets never took off in the way sun wanted them to. I am not quite sure why, but I think they lacked the simplicity that attracted people to flash.
Ajax has nothing to do with all this. Ajax is just a way to keep an entire page from refreshing by dividing it up into subsections that refresh independently. Again, this is me trying to explain this all as non-technically as possible.
EDIT: It seems I approached this answer the wrong way. To get alittle more technical; Flash is the most mature of the bunch. Silverlight and JavaFX are essentially babies, and while both Microsoft and Sun are trying to woo developers from their existing base (.net and java), I don't know if anyone can say anything definitive about either technology. It is going to take alittle while to see what technologies take off.