However, for a full explosion, you need a series of pictures, like a movie. Contrast this to the other approach, which requires almost no artwork, but just a good set of algorithms. We might try this method, but I'll need to conspire with real artists to get the artwork for it. Lastly, particle systems are the ubiquitous element that gives any graphics engine that certain je ne sais quoi . A good analogy might be that particle systems give to graphics engine what gratuitous French quotes and analogies give to otherwise bland technical documents.
Particles are a good example of billboarding.
In theory, particles are points. So, the way you render a particle is to render a billboarded triangle with a texture of a particle on it. The generic particle texture is a colored circle with everything else fully transparent, so you get a colored circle when you draw it. That's what Quake 2 did. Of course, particles aren't anything special if all they do is sit there.