How would you make dust effects like the ones frequently shown in this video?
Using Blender Internal, you could use particle emitters and Halos: Monster Movie
He says halo emitter, but I still don’t understand exactly what to do, or what that means.
You should read up in the Blender Manual about Particles and Emitters, and in the Materials section, about Halo materials.
A Particle Emitter emits particles (makes sense, eh?) according to the parameters you set for the Particle System. The kind of particle emitted can vary. For dust, a halo material is good because it is very fuzzy, somewhat volumetric, is easily made different colors and opacity, and can respond to forces like Wind. In Monster Movie I use halo particles for all the fine dust effects, and object particles for the more solid-looking debris at the end. All produced by Particle Emitters with different settings.