Unfortunately I cannot speak for IE6 or IE7 as I utterly detest Microsoft internet browsers.
However, if you are using Firefox you can make use of the Add-On known as [b]Media Pirate 2.3[/b].
Just grab it from the Firefox Add-Ons page, restart Firefox and a little globe with a downward-pointing arrow (greyed-out) will appear on your top menu bar.
Whenever you visit a site like YouTube, the greyed-out icon will light up in green and blue colours. Choose your video, click on the icon, a new browser page will open for Media Pirate, click on the Orange Downward-pointing arrow, click on 'Save' in the dialogue box, and watch the video-file download.
You can watch the streaming video while the video file downloads.
One word of advice: wherever you elect to save the video-file to, chosen when the dialogue box opens, the resulting saved video-file will be called [b]getvideo[/b] WITHOUT any DOT or extension.
Using the file manager of your choice, navigate to where you saved the file, right click it and 'rename' it to a suitable title with a DOT and the extension of [b]flv[/b]
The extension MUST be FLV so that it can be recognised as a Flash Video file. You can find programs to convert the file format if you need them converted.
Personally I just watch the video files with [b]GOMplayer[/b] which can handle FLV files by default. You can download it from here:
http://www.gomlab.com/eng/
Download, sit back and enjoy. :)