You really should read through the manuals for your engine but I will try to help you out.
http://www.lawnmowerforum.com/attac...14054d1367096263-wiring-help-please-image-jpg
In that picture, you see a soda-pop can sized object. That's the starter. See at the base a large red wire coming off? That is the positive lead that would go through the solenoid to start the engine. Negative is grounded to frame. On the right in that picture, you see a rectangle bolted right next to the starter, that's a rectifier. The yellow wires going back inside the engine go to the alternator. The red wire coming out of the rectifier is to charge the battery. Don't apply any voltage to those, you could fry either the rectifier and/or solenoid and they're not cheap.
http://www.lawnmowerforum.com/attac...14055d1367096820-wiring-help-please-image-jpg
In this picture I am guessing that the wire in the top of your hand (upper red one) is a kill switch for the ignition. I could be wrong so don't sue me
Another good reason to read the manual. The wire off to the right (yellow?) looks like a ground wire that was added by a previous owner.
I highly recommend going to the Briggs site and picking up the manual for your engine:
Manuals & Parts
Then, go to your equipment manufacturer (looks like MTD) and get a manual for the mower which will include a wiring diagram:
PDF Manual Web Archive