Window crank
Looks like the worm gear has stripped out. This cannot be repaired or replaced, unfortunately. You need an all new operator.
We can help, but we will need to see more of the operator itself. Specifically, I would like you to open the window as far as it can go and show me a clear, top-down view of the mechanism. That will allow me to match it up to something in our catalog.
It's not difficult, in my experience. The hardest part in your case may be due to the fact that you have a vinyl window. You need to find a way to get the vinyl sill trim up and off the operator so that you can access the screws, but that's tough because it differs from window to window. Sometimes it can be pried up, sometimes you have to take the frame apart.
If you aren't handy, or if you don't want to experiment, you may want to have a local professional come out and do it for you. Otherwise, as long as you can access the screws, installation is as simple as unscrewing the old one and screwing the new one on. The video below goes into more detail, though for a wood window.