Sometimes the only proper response to someone on Twitter, or even via Messages, is a GIF. With a proper search, you can find an animated image for almost any situation.
As I've mentioned in the past, managing GIFs in iOS is a pain. Boxie does provide for a simple method of organizing GIFs, but it doesn't help you discover GIFs.
Enter GIFWrapped. This free app not only makes it easy to sync and share GIFs using any one of your iOS devices, but it also gives you the ability to search (and then save) new GIFs. You can even connect your Twitter account to the app, making it easy to save GIFs your friends on Twitter have shared without having to do much work at all.
GIFWrapped uses Dropbox to sync your saved GIFs across your devices. Setup only requires you to link your Dropbox and Twitter accounts to GIFWrapped, and the app does the rest.
GIFWrapped is free, but ad supported. The ads can be removed through an in-app purchase for $1.99. I found during my time with the app the ads haven't been an issue, you may think otherwise after you begin using it.