Just download the app and sign in with your Amazon account. The Amazon Video app available for iPhones and iPads, Android devices, and Amazon's own Kindle Fires allows you to download videos to your device so you can watch them offline. Let's take a look at some of your options. This could save lots of valuable cellular data, too, especially if you're roaming internationally.
Not every service offers a download feature, but quite a few services allow you to download videos ahead of time so you can take them with you.