Another bit of info about the mystery of cover art:
iTunes is quite happy keeping cover art for your album in a separate metadata database, completely outside of your files.
I know this because my MP3s are stored as read only files on a Linux box, but iTunes happily assigns them album art when it has it.
Of course, when I do the command-I trick mentioned by Brad, the album art gets embedded in the MP3 file and iTunes uses that as the official one.
Note that iTunes will silently fail if the MP3 file is not writable and you edit the file properties. I have to make the files writable from my Linux box before applying album art changes (and this is intentional).
When you go to Amazon and find the artwork, you don’t need to copy/paste. Just drag the image right onto the Album Art square of the info page.