Barnes & Noble has released a version of its Nook Video for iOS devices. The app is fairly simple, letting you watch movies and TV shows that you've purchased from the Nook store. If you use the UltraViolet streaming service and have connected it to your Nook account, you can watch your UltraViolet movies through the app as well.
You cannot buy or rent videos directly from the app, as that would count as an in-app purchase. Barnes & Noble's Nook ebook reading software is similarly devoid of in-app purchases. Once you sync your video library to the app, however, you can can stream or download both owned and rented movies and shows, either over Wi-Fi or a cellular connection. When connected to the Internet, your playback position is saved, enabling you to continue watching a video anywhere. Nook Video does not support AirPlay, so if you want to watch your videos through the app on a TV, you'll need to use an HDMI cable. Nook Video is a free application, available on the App Store now.