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.
I haven't got much experience with different game engines but I imagine BGE is pretty hard to beat if one isn't going for the very best (commercial level) graphics. On the other hand with dynamic lighting, normal maps, video textures available in BGE I haven't thought of anything that I couldn't have.
One HAS to love the BGE workflow of being able to model, animate and script the game content as you go without exporting and importing all the time and hunting and fixing the annoying small problems and bugs that usually originate when different software interact.
I doubt anyone who says BGE is hard to learn or not user friendly. Traditionally one has to learn 1 game engine, 1 modeling/animation tool, a level editor and then find out how they interact. BGE has got it all and all the pieces are following the same logic in UI and keyboard shortcuts. The data is all there, at once.
And it's free. The GPL licence makes it very hard to sell but I suppose one could use it for game alphas (models, animations and environments can be easily exported if you want to go commercial and invest on a platform) or try the donation-based funding. After all, most of the effort is put (AI heavy genres aside) into modeling, animating, textures, sounds and level design, that is perfectly re-usable as they are.
In my opinion for the honest indie projects BGE should be among the top choices. But perhaps many people think they must learn to use the tools of the gaming industry because that's where they want to work in the future. Of course they have to be illegally downloaded or bought at ridiculous prices.