If you need audio, complete all these steps. If you want the whole video, u can kinda end at step 3.
Step 1: Copy the YouTube video URL. Try to search for the best quality version u can find.
Step 2: visit http://KeepVid.com where you can paste the URL into a bar and the website lets u download the file. I'd choose the highest resolution available.
Step 3: Get MPEG StreamClip. It's an audio/video converter/extractor/player. It does pretty much everything so it's just a good app to have. Also it's free and legal. Get it here: http://www.squared5.com/
Step 4: Open MPEG StreamClip and do File > Open Files, and locate the file u got from Step 2.
Step 5: As soon as u got it open, u can click play to make sure u opened the right file. Now go do File > Export Audio. Choose whatever format, change AIFF to MP3 or whatever if u want. (AIFF is higher quality and really should just be used if ur making a musical production or whatever.) Otherwise MP3 will be just fine. Make sure u name the file something u can remember.
You're done :-)