I can't seem to come up with what this "way of running/jumping" is called.
You start by standing straight, then bending the right leg, then jumping forward by using your right leg, then landing on your right leg first, followed by the left one. Then bending the left leg and then jumping forward by using your left leg, then landing on your left leg first, followed by the right one..
I used to do it a lot when I was a kid while playing, though I never knew what it was called.
Following the accepted answer, I found a good video showing the exercise(known as power skips) as I remember it.