I fell down the steps coughing causes me severe pain on my right side and back?

Almost certainly bruising to the intercostal musculature and the muscles of the back (depending on where the trauma occured). There's the possibility of a cracked rib from what you're saying, but injuries to these muscle groups are more common and have the exact same symptoms. If you feel there's a chance you've done some real damage, head for a doctor. However, if this injury is purely soft tissue (and even if it is a cracked rib), there's not a lot to be done for it except pain control.