"Renewed" is Amazon's marketing speak version of "certified pre-owned" basically. It's this thing where Amazon themselves take used phones and try to clean them up. These phones aren't handled by Apple or the originating companies in any way and they're not officially refurbished. Amazon just has their own in-house tech department clean the outside of the phone, install a fresh copy of the OS, and throw some cables in the box before they send it out. Maybe they'll replace the battery or screen if they know how.
For all intents and purposes the phone was fixed up by a random 3rd party street corner repair shop, and probably repaired using aftermarket unofficial parts. It's better than "used" because you'll at least have a guarantee that the phone turns on and boots, but anything past that can be a gamble. Reviews of the service are mixed, it's definitely one of those "you get what you pay for" type deals.
On the other hand, buying an official "refurbished" iPhone direct from Apple is near totally risk free. Apple replaces the entire outer shell of the phone, gives it a new legit Apple battery and stuff, and does a bunch of testing to make sure everything actually works right. They're indistinguishable from new iPhones except for the serial number. If you can find a decent deal for a refurbished phone on Apple's site I'd definitely suggest taking that option over an Amazon Renewed thing.
