I'm new to Adobe technologies and I'm interested in game development. In respect to this, what is the relationship between Flash, Flex, Adobe AIr, and ActionScript?
If you want to learn how the three different tools (Flex, AIR, Flash) are similar and differ you should search Google using terms like "Flex vs AIR vs Flash" - you will find folks have already tried to differentiate some combination of them. One thing they all have in common is Actionscript, though the extent/capabilities of the language between them varies.
Really? I found a few condensed explanations using those terms. If by "horse's mouth" you mean you want someone from Adobe to give them to you, you should search thru their web site in the products sections. These are user-to-user forums and Adobe personnel rarely visit here.
If you know a link that has what he wants, wouldn't it be prudent just to provide that?
I got overwhelmed trying to understand all of the different related elements of the Flash family when I was starting, so I know what that's like. Roughly (and Ned might want to correct or clarify some of this), Actionscript 3 is the programming language Flash/Flex/AIR content is based upon. Flash is a blanket term encompassing the developing environment (Flash Pro), code libraries, and runtime (Flash Player). AIR is a runtime environment built upon Flash Player, but with added functionality and cross-platform support. Flex is an extension of Flash, offering MXML components in addition to the libraries available in Flash Pro. Adobe intends for you to use Flex with Flash Builder, a separate IDE built upon Eclipse.
I haven't used MXML yet, but my impression is that (at least in Flash Builder) it is functionally similar to WYSIWYG editing in Microsoft's Visual Studio.
If you are developing games, you can edit entirely in Flash Pro, entirely in Flash Builder, or combine both of them. Flash Pro lets you put code on the timeline, which can make some coding much easier, but also makes it very, very easy to write messy and disorganized code. Flash Builder doesn't provide access to the timeline and is intended for class-based development. Personally, I like to combine the two, writing some timeline code in Pro and doing my class files in Builder, but I'm not the greatest developer.
As a heads-up, the code editor in Flash Pro is absolutely terrible, and is full of bugs that have existed for many generations, because Adobe wants you to buy Flash Builder for the additional $600, but if you are a student or "unemployed", you can get FB for free from Adobe.