First, find the YT video you wish to download. I shall use the amusing (And utterly terrifying) Pickle Surprise for my example, located at http://www.youtube.com/watch?v=NwEt6doEglA.
Next, look to the right of the video window and find where it says "URL" (Above "Embed" but below "Tags"). Directly to the right of that it will have a small box with an address in it. In the case of "Pickle Surprise," this address is http://www.youtube.com/watch?v=NwEt6doEglA
Copy the entire address.
Next, you have a choice. There are two websites that enable downloading of YT clips. Both do not always work. However, one usually does. They are http://keepvid.com/ and http://javimoya.com/blog/youtube_en.php Open one or both of them.
Paste the address you copied into the box below "Download videos DIRECT from most video sites" and above "example:"
Look to the right of that box. In the drop-down menu between the text box where you pasted your address and a button labeled "Download," find and click on "Youtube." It should be one of the uppermost choices. Note that many other sites also work with both of these.
Once done, click the "Download" button. Wait a moment while the site processes.
For Keepvid: once it is done processing, a link that says "›› Download Link ‹‹" will pop up in a box below the "Record," "Convert," and "Play" buttons.
For VideoDownloader (http://javimoya.com/blog/youtube_en.php): once it is done processing, a box will appear with a button that says "> Download Link."
For each of these, you have two options:
--Right-click on either the Keepvid link or the VideoDownloader button and select "Save Target As," "Save Link As," or whatever your browser users (All will have similar notation.
--Left-click on the Keepvid link or the VideoDownloader button.
A save window will pop up regardless of method. The only difference is that if you left-click, make sure you select save rather than open or open with.
Navigate to where you want to save the file.
Next, and this is important, rename the file to whatever you want MAKING SURE THAT THE FILE EXTENSION IS .flv! For instance, if it comes out like "get_video.htm" as VideoDownloader is wont to do (Keepvid appears to be attaching the .flv extension correctly at the moment), rename the file so that it has ".flv" instead of ".htm"
Click the "Save" button and allow the movie to download.
You now have the video file saved. However, you will need an FLV Player to view it (Although you should be able to open it in a browser and have it work.)
Go to Download.com and download the free FLV Player, located here: http://www.download.com/FLV-Player/3000-2139_4-10467081.html
Install the player. It gives you the option to associate it with any .flv files, meaning that if you were to double-click on an .flv file it would automatically open in the player. If you choose not to enable this, you can still open the player and navigate using its "Open" command to the file you wish to play.