|Give an example of a third party video please.|
Saving to HD is a whole big subject. There are various free "YouTube downloaders" around but they don't always work for other websites. I use "YTD Downloader" (free version) which covers quite a field and keeps updated and working when YouTube make one of their many website changes.
In IE you can sometimes get them out of Temporary Internet Files but you have to clear your cache immediately before download or there will be a lot to wade through - look for the big files. The freebie "VideoCacheView" can make life easier in IE. Whatever, you have to wait until they have played through for this to work.
In many cases they are temporarily cached in RAM rather than your HD, so there is no easy way around it. Firefox no longer uses a disk cache.
Always pop back and let us know the outcome - thanks