NOTE: AT&T names things where the numbering may be slightly confusing.
The Next 24, 24-month plan means that you are free to UPGRADE after 24-months, but you don't actually pay off the full price of the phone until 30 months. For example a $600 phone, on the 24-month plan, has payments of $20 a month. But $20 x 24 months only adds up to $480. At that point, you can trade in your phone for a new upgrade, and start a new payment plan for a new phone. Basically, they "buy" the phone back from you for $120, the difference between the original $600 you agreed to pay and the $480 you have already paid. However, if you choose not to "upgrade" at 24-months, you keep paying the original $20 a month for those remaining 6 months to pay that final $120. At that point, you owe no more money, and the phone is fully yours. No questions asked.
So, a Next 24 is actually spreading the costs over 30 months, but they let you sell it back to AT&T at 24-months if you agree to buy another phone from them.
You pay the full price for a phone.
Doing the math, though, Next is still cheaper than getting a subsidized phone under contract. Put it this way, your cellphone bill is going to be the same, but paying for it yourself, you add $20 to the cost of that bill. Buying it under contract adds $20 to the bill PLUS paying $200 upfront for the phone.
I am an old banker, so financing things is always a negative in my book. I've seen what unnecessary debt can do to folks, and I pay cash whenever possible. I resisted NEXT for a long time simply because I felt that "financing" a phone through a carrier was just a way for them to make extra money off me, and i'd rather just buy it outright. And if i'm buying it, I might as well subsidize it since I'll be on their service for longer than any 2-year contract they need me to agree to. But, after speaking to several with AT&T and doing the math over and over, and actually seeing the results in my bank account, NEXT comes out cheaper. Phone service providers are no longer giving away free phones, since smartphones already penetrate well over 80% of the market. Thus, having the customer buy the phone is a more attractive option for them, and it turns out better for you too these days.
TL/DR:
Next 12 = 20 monthly payments, with guaranteed trade-in at 12-months.
Next 18 = 24 monthly payments, with guaranteed trade-in at 18-months.
Next 24 = 30 monthly payments, with guaranteed trade-in at 24-months.
Next actually cheaper than subsidized contracts.