As you found an interesting section of Youtube video and thinking to turn that interesting video section into an animated gif picture file so that you can share that animating picture with your friends online or put it in your website, then you can try this website.
GifSoup.com can help you to turn a video into a GIF image but only limited to 15 seconds. This limitation is to avoid larger gif file which will take longer time to load. To create a gif file from Youtube, you will need to register for a GifSoup account first.
Sign in, paste a URL in the GifSoup website box, click the Create GIF yellow button. Later, set your start time and end time to preview. If you like it, click on finish and download to get your animating GIF image. Have a try!