My lawnmower will not stop running once it starts. I tightened the cable that runs from the handle to the engine, no luck. I saw a green wire that is loose - looks like it should be plugged into something. Could this be the problem - is this a ground wire? Do lawnmowers have ground wires?

Lawnmowers typically have a kill wire. When the kill wire is grounded the engine stops.

Look at the throttle linkage toward the SLOW side. The linkage usually makes contact with a metal clip. This is where the wire goes.

Yes, that could be the "ground" or Kill wire. Look at the throttle linkage close to the carb and see if there is a plastic block that has a small thin piece of metal that is straight and a piece shaped like a hump, then depress the straight piece, and insert the end of the wire, be sure the wire is stripped at the end.
In a pince and during troubleshooting, I have placed a plastic zip tie tightly around the boot of the spark plug wire to pull the long tail end of the tie in case I couldn't kill the engine. The zip tie works great and doesn't shock you.