Simple alpha blending and/or additive blending goes a long way :)

Found this by googling "alpha blending".

Really, it varies a lot depending on what it is you're trying to do. Some of it's postprocessing and fancy shaders, some of it's clever use of special models, other times it's just lots of alpha-blended quads everywhere :)

As a general rule of thumb, though, you're going to want to look into something called particle systems, which is the subset of computer graphics dedicated to rendering lots of small fancy things. If I may make a suggestion, I would advise grabbing a copy of the Unreal Development Kit and looking at all the neat little material tricks and particle controls in Unreal. Generally if you have some understanding of the 'how' of computer graphics, I would think that you could come up with a working system for actually implementing these effects in your project given the constraints on how they're made.