A lunge whip is supposed to be used as an aid when lunging a horse, which means the horse is on a very long lead rope and works in a big circle around the person - the idea is that because the horse is not next to the rider, the whip is used as an extension of your arm along with your voice to give the horse the signal to do things. It's a long solid stick with a long bit of rope or cord sticking out. If a horse has been trained properly, it shouldn't be afraid of the whip because in the right hands it's just a training aid.
So if you wanted the horse to go up a pace from walk to trot, you might flick the rope end of the whip behind the horse but nowhere near it, while you say your verbal command at the same time. The noise of the whip is part of the aid. You might use the very end of the whip to lightly touch the horse to get it to move further out, like a light poke - you wouldn't be close enough to use your hand to do it. Usually just holding the whip slightly behind the horses rear quarters to create a triangle between you, the front of the horse and the back of the horse is enough to let the horse know you want him moving forward.
There's never any excuse for what she did. Even though it's called a whip, you should never be actually whipping the horse with it.
You wouldn't ride with a lunge whip, but it would be pretty difficult to lunge without one and there's all sorts of reasons why you might want to lunge your horse rather than ride it - improving the bond with the horse, young horse, retraining the horse, horse or rider recovering from injury, wanting to exercise it, improving the horses balance, horse or pony too small or unfit to ride, improving the horse's confidence, training an inexperienced rider, improving the horses reactions to non ridden aids - just for a start.