Before discussing different ways to download video, let me first tell you that in most of the ways i am going to discuss, you will have to spend time watching that video online atleast once.
To download videos from youtube
For this, view the video online completely once, or atleast let it buffer completely while you spend time working on something else. Then go to the cache folder of your Browser and search for a file with .flv format. The name of the file might be anything. This is the video which you wanted to download. Copy it to any different folder before closing the Browser as the cache might be deleted after you close the Browser. You can view the .flv format video in any FLV Player or VLC.
To download videos from other sites
To download videos without watching it (It might not work on most of the sites)
For this, open the site from where the video can be watched online. Then look into the source of the webpage. If you are lucky, you might just find a complete link of a .flv format file. Copy and paste the link in a new tab in your Browser. The Browser will ask permissions to download the file. After the file is fully downloaded, you can enjoying watching it using any Flv player. This method works only on non-commercial webpages in which the user has shared the video for free for the public to watch it.
So now enjoy downloading videos.