I have balance issues, so I've fallen down the stairs many times, sadly. The worst time, as far as injuries go, was about sixteen years ago. I was carrying a heavy suitcase down some carpeted stairs and my feet slipped out from under me. I fell straight down, landed right on my butt, and snapped my tailbone into an L-shape. X-rays confirmed it was broken, but it never did heal right--causes me pain to this day.
Another time, I was carrying my 2-year-old sister down the stairs. I saw the landing, but misjudged the distance, thought it was only one step away, when it was actually two or three, so I took a wide step, expecting the landing to be under my foot. It wasn't. My entire focus was on protecting my sister, so I sort of cradled around her as we fell. She was fine, woke up scared, but uninjured. I turned my ankle. Nothing broken, just a sprain and some bruises.
As far as what it feels like, it usually happens too fast for you to think about anything while you're falling. Once you stop, you're often frozen for a minute, too afraid to move in case you're injured. There's this sick rush of nausea from adrenaline. It protects you from the worst of the pain, at least for a little while. Once the excitement ebbs, you'll start feeling jittery, as well as sore from the bruises and knocks. Even if you didn't damage anything, there's still pain, usually from grabbing for the railing and pulling a muscle or two, or just from tensing up as you fall and straining your back, or shoulders.