Apple
The "iPhone Upgrade Program" works just like the monthly installment plans you might be using with your carrier if you're not on a two-year contract. Here's how Apple's version works:
At a US Apple retail store, you can select your carrier and and buy a new unlocked iPhone, paying just the first installment. You'll pay the rest of your new iPhone's retail price in monthly installments.
The iPhone Update Program will let you pay off your new iPhone 6S starting at $32 per month, and it also includes AppleCare+, Apple's extended technical support program.
The monthly amount from the iPhone Upgrade Program will be added on to the monthly bill you receive from your carrier.
You might not see that much of a difference in your monthly bill with the iPhone Update Program, but it is essentially a more transparent way of showing you how much you're paying for your phone. Your monthly bill will also drop once you've paid off the full amount of your new iPhone.
The program is specially designed for those who don't want to lock themselves in a two-year contract with a carrier, which only allows customers to get a new iPhone every other year.
Apple
Most of the main US carriers are eliminating contracts entirely, so signing up for a new contract might not even be an option for many customers for much longer. At the moment, only AT&T continues to offer two-year contracts.
Some customers might be put off by the fact that they won't be able to buy new iPhones at steep discounts when they sign up for new two-year contracts. But the reality is that customers on two-year contracts were always paying the full retail price of their new phones, as the carriers would bake a monthly installment into their monthly bills to make up the difference of the discount.
The controversy of two-year contracts was that you didn't know exactly how much you were being charged to pay off your new phone, and if you kept your phone longer than the two years, you risked paying more than your phone's full retail price as your monthly bill wouldn't reduce to reflect that you've paid it off.