Starter won't engage when key is turned.

Great explanations above. I'll add:

The starter relay rarely fails.

The problem is usually the NSS. Not the part itself, but the transmission is not exactly in P or N due to a mis-adjusted linkage.

If you are sure the transmission is in P, connect a jumper wire from the NSS terminal to ground. Before you try starting, block the wheels and stand to the side in case you aren't really in P.

If the key still doesn't work, try by jumpering 12 V from the big stud to the "Start" terminal (yell wire).