It turns out most of what I have wanted to download is on YouTube and for that I use a free program called
YTD Video downloader (http://www.youtubedownloadersite.com).
I use the non-upgraded version. You can see the difference between the basic free version versus the $30/year ! version at
Contrary to its name, the documentation makes the claim "Download from YouTube, Facebook and 60+ popular sites". I don't know if that is indeed true, but it seems to have done the job for me in the past.
Frequently when looking for such things, I use http://download.com. They are a pretty good place to get reliable, free software that is free of bad stuff. Here is a list of the video downloaders that they use:
You might find something there if YTD Video Downloader doesn't work on a site.
Another very useful tool I always have around is VLC Media Player
This is an amazing program that seems to be able to play almost any video, and can convert from a video formatted in one representation to another representation.
Good luck and let me know if you find and really good downloaders.