Hi Ron:
YouTube videos are in 'flv' (Flash Video) format, so you'll need a media player which can handle the format. (Windows Media Player can't). Most people use VLC. It's free, here:
http://www.videolan.org/
Now you need to be able to capture the video to your PC. The most popular way to do it is to use this website:
http://keepvid.com/
There are two ways to use that site:
1. Copy and paste the URL of the required video into the address bar in the green box ;
OR
2. Drag the button (at the top of the page) onto your links toolbar. Then, when watching a video on YouTube, click on that button. You'll be taken to the KeepVid page, with the URL pre-loaded.
Then, whichever route you've used, click 'Download' and then click the new download link which will appear under the grey bar. (You might be asked if you want to open the file or save it. If so, select 'save to disk') You'll be invited to specify where you want to save the file. Navigate to a suitable folder. Change the default file name ('get_video') to something more memorable and add the '.flv' extension. (e.g. change 'get_video' to 'lionsleeps.flv'). Click 'Save'.
To view the video, open VLC. Select File > Quick Open File. Navigate to the folder which contains your video file and double-click on it.
Chris
PS: It's actually much, much easier than I've made it sound ;-)