We want our kids to watch some kinds of videos; for instance, a close-up of the sun. Amazing!
Other times, the kids just wish to entertain themselves, we lack the time to monitor, but we really wouldn't want them to start surfing freely amongst videos which may be highly inappropriate for their age (or at all, but that's a different problem).
There are several possible great solutions:
If you're OK with having the kids "loop-watch" videos (appropriate only) for a while, you can introduce them to Kideos - an excellent site, with age categories and theme channels; Totlol (requires a free sign-up); or Zuitube (turn on TV mode to watch all videos in the page). These three sites are mainly counting on Youtube for content, but each filters the clips to suit young kids, and disables the "related videos" Youtube feature.
PBS Kids' videos is another great choice, and so is KidMango, featuring cartoons and various animation episodes, which can be watched continuously.
If, on the other hand, you are looking for videos of a higher value, rather than a mere entertaining past-time, you will be more than pleased to discover The National Geographic Kids' videos. neoK12 contains "educational videos, lessons and games" (watch, for instance, this fascinating cosmic journey). Finally, BrainPOP is a great resource for animated videos in about any subject you can think of. I linked here to their page of free videos; the rest require purchasing a subscription (very much worth it, in my opinion).
What other websites can you recommend?