Its a development of .gif images and how we create it that becomes a great innovation to the digital photography. Firstly introduced by Jamie Beck and Kevin Burg who described in”an image that contains within itself a living moment that allows a glimpse of time to be experienced and preserved endlessly.”
It looks like movie file but its a .gif still image where some part of the images are moving smoothly. Since it was introduced, lots of photographers created this style, even amateur photographers can do it, here are mostly their first attempts to Cinemagraph:

Jamie Beck | Cinemagraph of NYC

Contemplation Cinemagraph | Jim Campbell

Aitor Guiscone

Simon Modery | My First Cinemagraph

Stepnanie Vu | My First Cinemagraph

RM Thomson | Jennifer - Cinemagraph

Unknown | Follow the link

Erik | American in Spain

Steve Mersel

Jaymi M | My First Cinemagraph