As PPs have said, don't look at AAA for the discounts. They do often have discounts of 5-20% at lots of touristy places, but that should be a perk, and not the primary reason for joining. To see how much it would save you on a WDW trip, call the CRO and ask what the AAA rate would be for the time you want to travel and the resort you want. Or talk to any of the Disney travel agents like Kingdom Konsultants, and ask for them to run the AAA numbers for you. The rate will vary by the season, and the resort. To simply use the AAA discount you do not need to book through AAA. Just like if you wanted to use the 10% at Hard Rock Cafe locations, AAA doesn't need to make your dining reservations, just show the card. The same concept applies for the WDW lodging discount.
Some of the discounts are national, and others are regional. If you go to AAA.com, and put in your zip code, the site it redirects you to should have a sections dedicated to discounts, so you can get an idea of what you would be eligible for.
I think the preferred parking is only if you get a diamond membership...i.e. the more expensive one.
OP: as others have said, this probably isn't worth it if you're just looking for a Disney discount. We booked our honeymoon through them in 2009 but got a discount through Disney, not AAA. Any discount Disney has available will almost surely be better. We did get other small perks - one free Photopass print and a free round of mini-golf I think - but we have AAA for when we need a tow, not for when we need a Disney trip booked.
The membership levels differ by the regional AAA you join. AAA itself is a national organization made up of local state and regional AAs. However, the preferred parking is something that most AAA travel agents will provide as a perk for booking through them. Other perks may include things such as free mini golf or pins. These will vary by office and the deal they have running at the time. This is just like how some Disney TAs will offer Disney gift cards if you book through them. Just an incentive to book a package through their TA service.
AAA is not worth it as a discount program. We have AAA and rarely if ever use it at Disney. You can book through them but it is not worth the cost unless you need AAA. We don't pay for it - my in-laws do. I don't know why as we pay for On-Star but they want us to have it so they pay for it. If they stop paying for it, I will not renew just for the Disney savings.
You can book through them (I believe) even if you are not a AAA member. The TA branch of AAA needs to make a profit just like their towing branch.
All in all, AAA is a good fallback discount. They are typically applicable year round, so you may be able to avoid paying rack rates during busy seasons. And if you do book initially using a AAA discount, you can normally rebook (pending availability) if a better discount is released that you are eligible for. If you are a reasonably savvy WDW tourist, you will often not want to book through AAA, but definitely try to use the discount through CRO. Their TAs are typically general TAs, and therefore know as much about WDW as they may about Rome or Tokyo. Some people do like their little perks like the fireworks viewing at MK, and the storytime with Belle (if they still even offer that) but I don't know if those are worth the hassle of having to educate the TA on what you want done and when it needs to be done.
