Since you say its not Boomerang than my guess is you're referring to Flixel. It's a software application that's a little costly but really cool. You can do the same effects in Photoshop or After Effects with a bit more effort though. That's how I make Cinemagraphs as they've come to be known.

On Instagram: https://www.instagram.com/flixelphotos/

Or Hashtag Flixel: https://www.instagram.com/explore/tags/flixel/

Hashtag Cineamgraph: https://www.instagram.com/explore/tags/cinemagraph/

Here's a tutorial to make it in Photoshop, there's lots of others: http://blog.spoongraphics.co.uk/tutorials/how-to-make-a-cool-cinemagraph-image-in-photoshop

Basically Flixel just helps you mask part of your video and loop it for you. I've done them in Photoshop and Premier. I find Premier to be the easiest. Just find a part you like, trim, duplicate, and reverse the time on the second part to -100 and you can get it pretty good with very little effort. Then can always save a frame and auto-import it to your scene and mask out parts you need to remain static.

I've run ad campaigns using them and absolutely see higher conversation rate from them.