When Klaus' mother (Esther, who created vampires) put a curse on Klaus to make his werewolf side dormant, she bound the curse using a girl who looks like Nina Dobrev. Then, in order to break the curse, आप would need that girl to pop up again in a few generations and use her blood. So she made the girl into a doppelganger, so that in a few generations she would have a descendant who essentially was the exact replica of her. So Katherine was one of those descendants and so is Elena, so they are essentially the same, kind of like identical twins, except Katherine is Elena's great great great grandmother या something. Idk, it's in one of the season 3 episodes.
