There are plenty of apps out there that let you make animated gifs in one way or another (for example: the fine Giffer) but I'm really taken with the results that come from Cinegram, as they are derived from short movie clips, not individual stop-motion animation style photos. It makes for a much cleaner and intriguing end result. This makes sense given that it's all a big take on the original Jamie Beck / Kevin Burg Cinemagraphs.

Aside from the actual images being made from Cinemagram, I can't really recommend it. It's obviously trying to cop the Instagram social network thing but it's all kinds of sloppy, confusing and poorly done. Just get it for the camera portion, don't even mess with the rest.