At this time, Amazon Prime Photos does not have an app — only web access with a desktop or laptop. You can however still access your photo library from other devices by logging into the Amazon website.
That said, there are some third-party alternatives for Android and iOS that allow you to access your Amazon Prime Photos library from your phone or tablet, with varying levels of features. Here are some of the more popular ones:
• Prime Photos from Amazon (Android): https://play.google.com/store/apps/details?id=com.amazon.photos&hl=en
• My Photo Stream (iOS): https://itunes.apple.com/us/app/my-photo-stream/id1137707638?mt=8
• Prime Photos Library (Android): https://play.google.com/store/apps/details?id=com.library.prime.photos&hl=en
• Prime Photos Explorer (Android): https://play.google.com/store/apps/details?id=com.photos.explorer.prime&hl=en
It looks like Amazon is working on an official Prime Photos app, which might be available sometime in the near future. The app would bring with it many features such as the ability to access and manage your photo library from your phone or tablet, edit photos, and share photos with others.
In the meantime, you can use one of the third-party alternative apps available to view, manage, and share your photo library from any device.