It follows the story of the Montagues and the Capulets, two families in Verona, Italy that have hated each other for generations. Romeo and Juliet meet and fall in love, but know that their families would never allow them to be together. Juliet takes a potion that makes her appear dead, so she and Romeo can escape, but he doesn't realize it's only temporary and kills himself rather than live without her. Juliet wakes and sees her dead lover and kills herself as well.
Their deaths make the two families realize how stupid the feud is and they decide to end it. This is a textbook example of a tragedy. Shakespeare never really believed in a Hollywood ending and killing off his main characters was pretty par for the course. I have read this story several times over the years both for personal enjoyment and as a requirement for various classes.
I think it's so popular because we have all had that one love that seemed doomed from the start. It could have been the high school romance that couldn't make it through the distance of college. The love who wanted a commitment, but you just couldn't do it. We all have had a love like Romeo and Juliet even if it didn't end in the same way.