I have a 35 Gtr and on the number 1 side there are three wires going into a box with a heat sink on it. Two of the wires run somewhere behind the flywheel and there is another red one which I believe is connected to the battery. Does anyone know what this box is and what the wires go to?

You talking about the voltage regulator on the #2 side (port side of boat)?

Regulator/rectifier. Takes ac current from stator and turns it to dc, then regulates the voltage, charging the batt.

That would be it. The connector shorted and I think it has been causing my motor to backfire. Why does the voltage need to be rectified?

Rectifier changes the output from ac to dc. Regulator keeps the voltage from being too high.

That won't make the motor backfire either.

I know what a rectifier is however I would have thought the current generated would have been direct. Anyway how would a poor connection there not cause the motor to shut off? I was stranded last weekend and motor would crank but not start but after playing with that connector it started working.

That connection is not your ignition. The only way I could think it would affect your engine was if it was shorting the ground on the coils maybe?? A lot of people cut those connectors off and hard wire it when they have probs.

So if I unplugged that connector my motor would still run just wouldn't charge my battery?

Yes

yes it will run if unplugged. no it won't charge unpluged. if the connector is melted replace it.most part stores sell a water tight wire plug connector. it won't run if its shorted. or it will be intermentent. lots of folks look to the rev limiter as the problem till they find the shorted plug. i like using mercury marine corrosion guard over dielectric grease from the simple fact that it won't melt under extreme temps. u can hard wire ur connections but. if down the road u want to check ur rectifer or stator u will have to disable the wires that u hard wired. here is a little advice to others every time u service ur engine u should always unplug the connector and check for corroded connectors and re-apply grease in the connector pins.