In a cocktail shaker, add the rum, lime juice, simple syrup, and mint leaves. Muddle (smash) the mint leaves well. Add ice and shake the cocktail, then strain into a tall glass such as a Collins or Highball glass. Add ice and top off with the club soda. Garnish with a mint sprig.
You can use any rum but white rum is traditional. And you can use plain club soda instead of lime flavored, but I think lime flavored is just a little better.