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.
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*