Casement windows are usually double windows that open from the center by turning the crank assembly. The assembly operates the extension arm attached to the crank. Each window portion has its own crank assembly. Without care and maintenance, the cranks can stick or break off.
The assembly works with the use of gears. When you turn the crank on the window, it turns a gear inside the gearbox installed on the underside of the window. The teeth of this gear mesh with a second gear to move the end of the extension arm along the base of the window sash to push open the window. Reversing the direction of the cranking pulls the window closed. A separate lock for the sash keeps the windows tightly closed and locked into place.
Regular cleaning and oiling of your casement window's crank assembly can prevent problems. Open the window and pull out the extension arm. Wipe out the track the extension arm follows, using a brush. Lubricate the moving parts of the casement assembly with spray lubricant or mineral oil, then reassemble the extension arm. If the gears in the assembly don't properly mesh or if the crank has broken, you'll need to replace the assembly.
Open the window to remove the end of the extension arm from the base of the window. Unscrew the gearbox from the windowsill. Note whether the window opens to the left or right and purchase a replacement accordingly. Install the new casement crank assembly reversing the steps you followed to take the old assembly out of the window.