You can rent for as many days as you need. You get a 10% discount if you do multi-day and prepay it. You go up to the cashier and he/she will give you a string of tickets, one for each day.
The great thing about the multi-day is that you can bypass the line to pay for a stroller and go straight to the pickup line. Just present your ticket, they'll stamp the date on it, and pick up a stroller. Saves so much time doing it that way.
If you picked up a stroller that day at one park and you decide to park hop to another park, you can still get another stroller at the 2nd park. Just show your multi-day ticket (it has the receipt stapled to it), your name tag, and pick up another stroller.
Each park has tons of strollers. Even during the busy times I've never had a problem getting a stroller, even later in the day.
Then when you're done for the day, you can still get $1 back. Just bring your ticket/receipt back to the cashier and they'll remove a stub from your ticket. Then you get your $1 back in Disney Dollar.
I love the multi-day stroller rent - my kids are 7 and 4 and don't really need a stroller but we usually get one when we do Epcot and AK.