Many people confuse moving their media folder with moving their library. A library is everything inside the iTunes folder. Unless you have a specific reason for not wanting to move the entire library it is best to do that rather than just media.
What are the iTunes library files? - http://support.apple.com/kb/HT1660
More on iTunes library files and what they do - http://en.wikipedia.org/wiki/ITunes#Media_management
What are all those iTunes files? - http://www.macworld.com/article/139974/2009/04/itunes_files.html
Where are my iTunes files located? - http://support.apple.com/kb/ht1391
iTunes 9 [and later]: Understanding iTunes Media Organization - http://support.apple.com/kb/ht3847 - plus supplemental information about organizing to new structure https://discussions.apple.com/thread/6477809?answerId=26404702022#26404702022
Image of folder structure and explanation of different iTunes versions (turingtest2 post) - https://discussions.apple.com/docs/DOC-7392 and making an iTunes library portable.
If you move only media then the main library files remain on the computer. If you want to move the whole library:
Quick answer if you use iTunes' default preferences settings: Copy the entire iTunes folder (and in doing so all its subfolders and files) intact to the other drive. Open iTunes and immediately hold down the option (alt) key (shift on Windows) so you get a prompt to choose a library, then select the copied iTunes folder.
If this is to a new computer and you put the copied iTunes folder in the default location of Macintosh HD > Users > *User Name* > Music then you don't even need to start with the option key held down, iTunes will automatically look for it there. (Make sure there isn't anything already in the iTunes folder there that you want to keep since you will be replacing it with the one you are moving.)
