We use "fell down" to refer to having failed in something OR collapsed.
(Example: I fell down on my History paper.)
(Example: I fell down (collapsed and dropped to the ground) when I was running and cut myself.)
We use "fell over" to refer to something that toppled and fell down OR tripped over something.
(Example: I fell over a rock (tripped over a rock) when I was running and cut myself.)
(Example: The tree fell over and crushed the car.)
Hope this helps.