Cool, you're an AS3 developer.
Alright, imagine ActionScript without Flash Player. It's kind of weird, but what if you had ActionScript but could not create Sprites or MovieClips, Sounds or URLRequests? You'd have the ability to write code, to make classes, to compile (probably) a command-line process, but you wouldn't build a game that way.
That's a little like what you get using Haxe -- a cross-platform programming language, but it does not have graphics, sound, events, network requests and most of the other things that come stock with Flash.
NME extends Haxe, and provides cross-platform build tools for targeting desktop and mobile platforms. It takes the "magic" cross-platform ability of the Haxe library and applies it to supporting Windows, Mac, Linux, iOS, Android, BlackBerry, webOS, Flash and HTML5 using the Flash-style API.
I would recommend you give NME a try. If you like the experience, you might also discover some of the other ways you can use the Haxe language (programming server-side code, writing strongly-typed Javascript and other cool uses), but NME will be the closest parallel to your experience as a Flash/AS3 developer.
I apologize how the namespace feels crowded.
NME uses Haxe typedefs to redirect classes, depending on your target platform:
package nme.display;
#if (cpp || neko)
typedef Sprite = native.display.Sprite;
#elseif js
typedef Sprite = browser.display.Sprite;
#else
typedef Sprite = flash.display.Sprite;
#end
When you use the nme.* classes, it automatically becomes one of the other classes, based on your target platform. I hope that makes sense? Actionscript does not support conditional compilation (#if #else #end) or typedefs, but those are two of the features which makes something like NME possible. You target a single API, but the underlying implementation changes drastically depending on whether you going to Flash, C++, or Javascript.