If you're on Windows XP and running Firefox, here's how you can locate a YouTube video in your browser cache, just follow the steps:
1) Turn on the option that allows you to view hidden files on your system.
2) Follow this path: C:\Documents and Settings\Username\Local Settings\Application Data\Mozilla\Firefox\Profiles\abcdefgh.default\Cache
Here 'Username' is generally your current account in XP, and 'abcdefgh' is some random 8-digit filename.
3) Once in the Cache folder, turn on the 'Detailed' file view option from the View tab.
4) Arrange icons by size or date, whichever you prefer.
5) Generally the YouTube videos will be considerably larger than other files present in the folder. By selecting a file that was downloaded around the time you were watching the video, and copying it elsewhere and changing its extension to '.flv'... you can view this video in any video player that supports flv files.
Enjoy.