What are possible back injuries from falling down stairs?

Many. Trauma such as a tumble down stairs can fracture the spine, inflame the joints or rupture a disc. Symptoms may be localized back pain or radiculopathy leg or arm pain, you should have this checked out by your physician.
See a Dr. If you have low bone density, can be at increased risk for compression fractures. Trauma like that can result in other fractures, disc herniations, pinched nerves (if pain radiates into the legs/arms), aggravation of underlying arthritis, soft tissue contusion (bruising), etc. Should see a dr, they may get an imaging study depending on your symptoms and exam.