The latest trend for web sites is a 'cinemagraph'. This is like a high quality gif. Let me explain;
A GIF is a short image sequence - usually designed to be informative or funny, but not necessarily aesthitic or high quality.
GIFS are very popular because they provide visual interest where an otherwise static image could only be possible. This is great in things like social media posts, emails and even on websites.
A Cinemagraph is in some ways similar. It is created by taking a short video sequence, a single frame of which is used as the static image overlaying the video, and a portion of that image is removed to allow the video beneath to show through. This is then rendered so that the file size is small - either as a gif format or a video file.
The result - when done correctly - can be stunning and visually engaging.
The procedure is of course not as easy as all that - it moves more into the field of an art form rather than a simple communication.
The video sequence has to be correctly prepared - it needs to be taken on a tripod, and the still and moving parts of the image should be carefully considered before hand.
The sequence to be used has to be correctly chosen, and then an artists approach is used to determine the areas to use as motion. A subtle approach seems to the best.
Here is a great example from the website flixel.com: