Although much more famous for his work on monster and fantasy movies, Ray Harryhausen also produced a series of shorts based upon fairytales during the 1940s and '50s. These films included stop-motion animation versions of Rapunzel, Hansel and Gretel and of course, the above Little Red Riding Hood.
I have to admit to finding Harryhausen's version of LRRH a little odd, though. The narration seems to add little that isn't communicated by the animation itself and the whole story is strangely bowdlerised, with no-one getting harmed, eaten or killed (which raises the question of why the wolf doesn't just come back and get Red Riding Hood the next day, I think).
This video comes courtesy of Cometin-Jetson who has also posted the other Harryhausen works mentioned above.