Please help! What is the name of that kidney disease where the urine backs up into the kindeys?

Reflux. I think what you are talking about is vesicoureteral reflux (vur). This is more commonly seen in children, especially males. There is a secondary form that can be seen in adults or anyone with damaged nerves to bladder resulting in a neurogenic bladder. When the urine backs up into the kidneys hydronephrosis (swollen kidneys) can develop.