That being said, the way I recommend to download YouTube videos is:
- find the video
- copy the ULR (uniform resource locator, aka the location) from the address bar of your web browser; probably the easiest way is the right click it and choose copy
- past the URL into the "Enter YouTube URL:" box on TechCrunch's Video Download Tool and click Get Video
- if you are using Internet Explorer, you'll probably see a yellow bar come up at the top of the page... click that to download the file
- you should now have the video file saved to where you had specified
Now, the video is in Flash video format, so you need some player like VLC to play it. If you are not able to install VLC because you are not the administrator of your computer, you can use the portable version.
That's it in a nutshell, comment if there's something I'm missing.