Trying to understand Bayesian reasoning about probabilities

I'm trying to understand what 'Bayesian' analysis of probability is. It's been explained to me in conversation several times, and I've just read an explanation of it in a book on probability ("Chances Are"), but none of these has resulted in any learning or understanding on my part. Of course this is an excellent illustration of what I'm always saying from my perspective as a teacher - if you don't actively work with the concepts you won't learn them.

Part of the problem is that, although explanations start out in plain English and simple arithmetic, they soon lapse into statistical symbols and equations where everything is prefaced by 'P', causing my reading style to switch into 'skip over this dreary bit' mode.

I do think it's probably important that I gain some understanding of this new Bayesian stuff, because it appears to be very popular among people who know what they're doing with probabilities. So trying to explain it in the blog is a way to force myself to figure it out.

Now I'm reading an on-line 'intuitive' explanation by Eliezer Yudlowsky. He emphasizes intuitive and visual explanations (and how the way that a problem is presented affects our ability to understand it), and so far it makes sense. He also has interactive applets where the reader can change the proportions and probabilities - another way to minimize passive reading.

So far I'm about 40% through his long web page, and I'm not sure I've learned anything yet. Maybe tomorrow.