Real Madrid's closest rival, Atletico Madrid, was founded one year after Real and has been one of their only real competitions in the La Liga historically. They play two intense matches every year, and Atletico broke Real's 8 year undefeated streak in the League. Real leads the series with 107 wins and 51 losses.
Madrid's biggest competition is Barcelona. Over the last fifty years, they have typically been two of the best teams in the world, and they're in the same league. Barcelona has one of the best records against Madrid, but Madrid leads the series with 81 victories and 76 losses.