I would like to know how to download a live video from the internet,
The Video DownloadHelper add-on does work for both standard video files and Flash. On Windows, I use Internet Download Manager to download all video (it has browser integration). On Mac I use video DownloadHelper.
The above method only works for non-live video., e.g. YouTube. You can't really download live streaming content as the entire file doesn't exist. You can only capture what is coming to your computer.
I used one of the tools listed here before – can't remember which one: How to download streaming video and audio
Or have a look here.
Try Firefox extension Flashgot. It'll notify you when there's some video or music on the webpage.
Or you can try Firebug. It's a debugging tool but maybe the best tool to download video. After installing Firebug, press F12 to open it and active "Net" panel. Then go to your video page. You'll see requests logged in Firebug. Scroll down to find the video file (you can see its extension, or if it's still loading and taking a long time to download) and copy the request location to a download manager.
This may not work if the site checks cookie before streaming video.