Former AT&T Employee here:
AT&T Next is pretty much 20 or 24 month interest-free financing. You walk out of the store with the phone for your states/county's sales tax %, and can pay off the phone literally anytime. NOTE: people on old school "combined billing" will NOT be able to pay the phone off early. (I just left the company recently, checked back with a friend and she said this still holds true. Had this issue a few weeks ago with a customer trying to pay their Next off and they couldn't because of combined billing. So until they change this, keep it in mind.)
Next does NOT affect your plan at ALL! But if you are on Mobile Share Value, you retain your $15 credit (300mb - 6GB plans) and $25 credit (10GB+ plans) if you pay "full price" for your phone(AKA AT&T Next). You do NOT need to trade it in to upgrade again. Once the phone is paid in full, you can do another Next. Keep in mind, if you do want to trade in, and upgrade at 12 or 18 payments you can do so ANY time. You just have to give AT&T 12 or 18 months payments UPFRONT + your phone and boom, you can start another Next. Or just wait the 12 or 18 months. Whenever AT&T gets their money pretty much. Though, to do this, you start a new installment all over again.
EDIT: As of right now, you can only do 4 installments of Next on your BAN (billing account number). This means if you have a family of 5, 1 person won't be able to do Next. This counts for any tablets you are paying off on Next as well.
AT&T will also let ANYONE who has 18 months or less (YES, one and a half years) left on their existing two year contract start doing Next as long as you have't missed any payments or owe them money recently. (Check with your local store on this one, but the company is letting people out of their 2 years VERY early). This will let you out of your contract with out a termination fee and then you'll be on Next.
Hope this helps. Feel free to post this in other threads talking about AT&T Next. I always hated seeing people confused about Next because it's so much different than what we're used to, and you get a different answer out of every rep you speak with.