"Fall over" means your body hits the ground after slipping or tripping. "Fall down" can be used for exactly the same thing, but it can also be used more literally, such as "to fall down a hole".

Fall over is more of a figure of speech. To "fall over" (oneself) - to go out of your way to do something. "Fall down" is used literally, in that your body actually hits the ground, usually unintentionally.