Plenty of references in Tolkien’s foreword to the Ballantine paperback, as well as on the copyright pages and front cover, imply that an unauthorized paperback version of LotR was printed. What’s the story behind this?
The full story:
LOTR was originally published in England. The British publishers wanted to put out a first-class hardcover edition in the U.S. Donald Wollheim of Ace Books offered to do it in paperback, which they refused.
However, the copyright law at the time stated that you lost copyright if a certain number of a book published outside the U.S. was imported into the U.S. Wollheim realized that some copies of LOTR had been imported and the number was large enough to void the copyright in the U.S.
The book was first published in the U.S. by Ace in paperback (I happen to have the copies – bought used, so I’m not a party to this 
). This was probably around 1964 or 64 – my copies don’t have a copyright date on them.
Ace paid not royalties to Tolkien. To fight for this, the newly established Science Fiction Writers of America made Tolkien an honorary member (since he hadn’t been published officially in the U.S.) and went to bat. Wollheim eventually paid royalties, claiming he always meant to pay, but couldn’t find Tolkien’s address. Right.
Eventually Tolkien rewrote the books just enough (adding the material at the end, for instance) for it to get its copyright restored. Ballentine Books put out their authorized edition, with the warning from Tolkien to buy no other. Eventually, the Ace editions went out of print. They pretty much had vanished from the scene when I first saw LOTR in paperback around 1968 or so.
I enjoyed the ‘Note on the Text’ section on LOTR regarding the prints of “The Fellowship of the Ring”, published on October 21st, 1954 by Houghton Mifflin Company of Boston, MA. Apparantly they tried to correct many errors they percieved in the text. Corrected ‘errors’ included dwarves to drarfs, elvish to elfish, further to farther, nasturtians, nasturtiums, try and go to try to say, and gasp elven to elfin.