Amazon has released an app for Prime Instant Video for the iPhone and iPod Touch, but it is strangely lacking Airplay compatibility. An app for the iPad has been available for months, so it's not clear what took Amazon so long to migrate that app down to the smaller-screened iPhone and iPod Touch.
It's pretty similar to the iPad app in that you get access to hundreds of videos from Amazon Prime's video library. You can also watch rented and purchased videos from within the app and download videos to watch later when you might not have access to an internet connection, or only a limited cellular connection.
To buy videos, however, you'll have to head to Amazon's website, purchase the video, and then return to the app. VentureBeat makes the point that this is probably because Apple takes a 30 percent cut of all in-app sales. The app is available for free through the App Store, though you'll need to be an Amazon Prime member to use it.