Ben E. King: Stand By Me
This is a great song, and a touching one that also manages to keep a good beat and be fun to sing along with. All winning elements that helped it get into the top 10 charts, not once, but twice, decades apart.
The song was released in 1961 and hit #1 on the charts, but then in 1986 a movie by the same title was released and renewed interest in the song getting back into the top 10 once again. Admittedly, that's when the song made an impression on me. Of course, I was only 7 years old at the time, and wasn't allowed to see the movie when it came out, but the song was on the radio, and I loved it.
Despite the song being pretty much the theme song for the movie (though it doesn't appear until the credits at the end), it was released in 1961 and the movie takes place in 1959.