The Blender Game Engine is a component of Blender, a free and open-source comprehensive 3D production suite, used for making real-time interactive content. The game engine was written from scratch in C++ as a mostly independent component, and includes support for features such as Python scripting and OpenAL 3D sound.
I have used before many engines, such as Dark Basic, game maker, rpg maker, Blitz3D and Panda3D. The Blender game engine has an excellent workflow(inside the 3d program), has a easy, acceptable fast(not fast as C++ thought) and largely supported programming language(python); is free; packed with features; is still actively maintained(more by the community than blender.org devs thought); has many kind users and exports the game to Windows, Linux and Mac, and maybe in the future Android.
I constantly check if I should use a different engine, and I always end up seeing which the BGE is still fit for my needs.
The BGE is actually quite good - the advantages of being easy to code in and having possibly the best workflow in a 3D game enine far outweigh the fact that it isn't extremely fast or powerful. The fact that it is free and open-source also makes up for any lacks in its capability.
2/14/11 EDIT: @AndrewIsGreen - The screenshots are fooling you. Check out the newest images uploaded.
Having used it for 3 years, I very much recommend it! Between its ability to support a project's entire workflow, including a built-in game engine, to its support for Python scripting, support for 3rd-party plugins, cross platform functionality and generous licensing (completely FREE!), there is little more one could ask for. Add to that an active community and endless resources readily found on the web and it's a no brainer go-to solution.
Besides the fact of the Open Source and public domain (: . The way blender mix a whole bunch of tools allow us to make everything we want. But the thing I love the most, is for now so basic that maybe no many people thing about that; but the way Blender Render works with BGE its just awesome!
*my bad, for the poor english*