Once I had a piece of paper
It was supposed to mean that you were mine
It looked like any other piece of paper...
It's kind of a depressing topic for a song. The guy's getting a divorce. The pieces of paper are marriage and divorce documents, all that's left of his union. Just between you and me, it doesn't sound like he's too cut up about the whole situation. The words indicate that perhaps he is not the most introspective individual, while the music features what sounds like a sitar, of all things.
Like "MacArthur Park," which was the song that compelled us to listen to this record tonight, "Just Another Piece of Paper" was written by the prolific and fabulous Jimmy Webb. If you have not heard his interview on World Cafe, you really should queue it up sometime soon. He wrote "Up Up and Away!" (And "MacArthur Park?" AND "Just Another Piece of Paper?" Prolific, fabulous, and...varied in quality.)
*Obviously, only the Richard Harris version will ever do.