I'm pretty sure they're just two names for the same thing... Chi dori means "Thousand birds" and rai kiri means "Lightning Edge". I think Chidori was the original name (because the attack sounds like birds chirping), but the legend is that Kakashi used it to cut lightning - whatever that means - so it acquired the nickname Raikiri.

Long story short, they're the same attack - the one where lightning gathers in your fist and you use it to punch a big hole in something/someone