What is the major difference between the A, S and P line of cameras?

Good question!

The A-Series PowerShots all use AA batteries. That's where the "A" designation comes from. They also have a full range of exposure controls so they're a sort of compact do-it-all camera. I often describe them as an inexpensive, less sexy version of the G-Series cameras. If you compare the features, the A-Series Canons are very impressive.

There really isn't an S series of Canon compact digital cameras anymore. There used to be an S400, S700, and the last was the S800. These cameras were compact versions of the G-Series cameras. They had the same sensors as the corresponding G model, but a smaller body, no hotshoe, RAW capability, or roatating LCD. Now there's an "SD" series of cameras. Those are also called "Digital ELPH" cameras. They're pocket-sized point-and shoot digital cameras with pretty simple controls.

Canon doesn't have a P series of cameras and I don't think they ever did. Sony did, although I believe that line has been discontinued. They were Sony's simple, pocket point-and-shoot digital cameras.

Does that answer the question for you? Let me know if you have more questions.

...There really isn't an S series of Canon compact digital cameras anymore....

I guess when I read "S" I thought it was synonymous with "SD", no? What about the S3 IS and S5 IS? Or do you mean they are not "compact"? They do, however, use "AA" batteries.