I don't know how verizon works but with sprint you have the option to do 18 month leases or 24 month buy out. With the lead, it's usually a little more expensive. After 12 months, you can trade in your phone for a new one of your choosing. At that time, you will start monthly installments based on the new phone. With the leasing option, you can trade in the phone to get a new one or buy it at the end of the lease to keep it.
With the 24 month installment plan, you pay a certain amount each month so that at the end of 2 years, you end up keeping the phone. If at any point in time you want to upgrade to a newer phone, you must pay off the remaining balance of your phone. You will be able to keep it once you do that. In some cases, you can choose to trade in the phone after 12 months (without paying for the phone or keeping it)