If you want to reduce the size of gif images gfycat is a free web service that will help you to reduce the size of gif images. It will reduce the size of Gif smaller than HTML5 videos. No limit for file size.
To make reduce size simply upload gif image into gfycat after some process you will get reduced gif images. These images can be pause, slow it down, speed it up, step frame by frame, or play in reverse. Gfycat provide embed code for reduced images then you can embed reduce images into your website.
Gfycat will help you to convert your youtube videos into Gif images. It supports top video hosting sites it includes YouTube, Vimeo, and Liveleak.