BDX is a small 3D game engine, designed to work with Blender (as an addon), and built to leverage libGDX, along with the Bullet physics library (specifically, the gbullet port). It features a straightforward execution model, supported by a relatively simple API, enabling the programmer to effectively define object behavior within a typical blender scene.
From a "build and deploy" perspective, BDX projects are simply extended libGDX projects, which means that you can use the same gradle-powered pipeline to deploy BDX games to any platform that's currently supported by libGDX (Windows, Linux, Mac OS X, HTML5, Android and iOS).
BDX is licensed under the permissive Apache 2 License (just like libGDX): You can use this software free of charge, no strings attached, in commercial and non-commercial projects.
For game development with BDX, you only need a relatively recent version of the Java Development Kit (I tested with 1.7).
If you plan on developing the engine itself, you'll additionally need a working version of ant.
Download and Install
You can find the latest release here.
Download bdx.zip. Once you have it, you can install it like any other blender addon - In User Preferences, click on the "Install from File" button, then find and install bdx.zip.
After that, you just need to enable it (you can find it in the Import-Export category - Testing support level), and then Save User Settings, so you don't have to re-enable after every blender restart.
At that point, there should be a BDX panel in the Render properties window.
You can find relevant information in the wiki.