Well, purely for educational purposes obviously, I'll tell you about stream ripping:
On your computer, go to a Youtube video of a song you want to obtain. Right-click in your browser's address bar to highlight the URL and then click 'Copy'. Go to an online file conversion site, such as this one:
http://www.flvto.biz/
Right-click in the blank field and select 'Paste'.
Check that 'MP3' is selected and click on 'Convert To'
When your file has been converted, click on 'Download' and select the location for your file.
Wait for the download to complete
Job done.
You can then transfer your collection of files across to your phone.
Note 1: In theory you can do the same thing directly on your phone but it's sometimes harder to do things like copying and pasting URLs on an Android (or other mobile) device.
Note 2: There are loads of file conversion sites like that one. As Youtube keep tweaking their system though, sometimes you'll find that a particular site doesn't work. If so, simply google 'Youtube mp3' to find another one. (I know that one is currently working because I've just tried it).