"Cry to me" is widely known because of its inclusion in a movie called "Dirty Dancing". I never saw that film. I came to know the song by the version recorded in 1965 by the Rolling Stones with kind support by Jack Nitzsche. It was written by Bert Russell (Bertrand Russell Berns - he also wrote songs like "Under the Boardwalk", "Twist and Shout" and "Hang on Sloopy") and first recorded by Solomon Burke in 1962 (or 1961, sources differ). The Solomon Burke version was arranged by Klaus Ogerman from Germany. Some sources have it, that Ogerman wrote the song, and Berns arranged it. But normally, credits for songwriting are given to Russell on the records featuring the song. There are many versions of it, f.e. by The Pretty Things or Betty Harris (also arranged by Bert Berns). Here is the "original" recording by Solomon Burke, the ultra-cool rendering by Professor Longhair from New Orleans and a very fine "Bayou-style" version by Marc Broussard featuring his father Ted Broussard on electric guitar.
Marc Broussard and his Dad: