UPDATE:SEPT 29/16 : I now just use www.clipconverter.cc. Paste the link to the YouTube video, choose your format and download the video. Works great for music as well.
There is a easy way to save YouTube videos.
While the video is playing, simply right click anywhere in the player window and select POP OUT.
The video will start playing in a new window. Right click anywhere in THAT window, and one of the options is "save video as".
The video will be in MP4 format, although it won't come up with that file extension on its own. It wasn't going to save the video with any file extension, so I ran it through mediainfo and it said MPEG-4. I saved the file with an ".mp4" extension (by typing that after the file name I chose) and I was able to play it just fine in my VLC player.
Please keep in mind though, this method only works with files that are not copyright protected by the user who uploaded it.
Another method is to simply add the letters "ss" to the url in the url bar at the top of the page, right before the word "youtube".
That means that "www.youtube.com/watch?blahblah" becomes "www.ssyoutube.com/watch?blahblah".
After typing in the "ss" hit enter, and it will take take you to savefrom.net, where the youtube file will now be available in links at the right side, in many different types of file formats.
I haven't test this way much, but I believe it will let you save any video, regardless of the copyright status.