As the author of that other post, I'm happy to see Terry Tao's post on the front page now too. I learned a lot from it when I was researching for my own. The two articles end up going in somewhat different directions---Terry doesn't talk about dimensions with complex exponents, for example---but his is definitely more complete.

My favorite bit is his explanation of how dimensional analysis explains the relationships between powers used in Sobolev inequalities. After we get the idea of dimensions divorced from the usual physical MLT usage you can apply these techniques all over math.