I usually do it the "ghetto" way.
First download a screen recording program, then start the DVD so the scene you want to record is ready to play. Pause it at that scene. Begin recording with the screen recording program, play the DVD from where you paused it, and when the scene is finished, stop recording.
Obviously, you'll have a video file that begins too early and ends too late (unless you can perfectly time it), so you'll need to take it into a video editing application and shorten and/or zoom in/out of the video so that you get only the section you want people to see and not the entire view of your desktop, for example.
Now, once you've done the necessary editing, you can export it as a .avi file and use that to turn it into an animated GIF. There are other ways of doing this, too. One of the applications I use (Movie Edit Pro) has a built-in ability to create GIFs from the movies you edit without having to use any external applications. That's probably one of the best features ever if you're into creating sweet GIFs from just about everything you can think of!
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