If you are regular user of twitter, then you must observed moving pictures in the display image of some folks. Such type of images are known as animated GIF because of format. Recently after the huge change in profile layout, release of Filter apps for Android and iOS devices, Google + also introduced an option to add animations in your profile picture.
If we look towards mainstream social networks like Facebook and Twitter. Facebook still not in favor of
such images and not supporting them, well few days ago Twitter also decided to block this feature. People are showing different opinions on this announcement; according to some of them it sucks internet speed and increase time to download a page, well some are in favor because it is helpful to get an attention of online community.
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You can create your Animated Gifs easily either via Photoshop or can also download from internet; but online one are usually watermarked. You can use it as your profile picture, on both desktop and mobile.