The first championship was Super Bowl VII when the Dolphins defeated the Washington Redskins by a score of 14-7, which marked the NFL's only undefeated season (1972). The second championship came the next season in Super Bowl VIII as the Dolphins defeated the Minnesota Vikings 24-7.
Outside of the 2 victories in Super Bowl 7 & 8, the Dolphins played in Super Bowls:
VI - lost to to the Dallas Cowboys 3-24
XVII - lost to the Washington Redskins 17-27
XIX - lost to the San Francisco 49ers 16-38