When you turn the key to the on position you should hear a faint click, which is the carb solenoid opening.
If not then locate the fuse which is usually on the right side of the mower behind the dash / turret driver rtation, whatever you want to call the bit that the steering wheel pokes out of.
It is a std auto blade fuse.
Replace it, even if it looks good ( use a spare from your car ).
IF the engine still does not crank then locate the solenoid , it should look like this
http://jdpc.deere.com/jdpc/servlet/com.deere.u90490.partscatalog.view.servlets.HomePageServlet_Alt
the bolt on terminal have wires that go to the battery and to the starter.
The one that comes from the battery should also have a thin wire coming off it ( the fuse will be connected to that wire ).
Jump that wire to one of the push on terminals near the bottom, Jump the other to ground. Polarity is not important either wire will do .
That should cause the solenoid to activate and the engine to turn .
If that does not happen then jump directly from the battery - to a good ground near the starter and the battery + to the + wire on the starter.
You will get a big loud spark if the battery is good and the engine should turn.