The headlights on my '77 Jeep J10 go off and on. Can anyone tell me where I should begin to look first to track this problem down?

Is the headlight switch turned on or off when this happens?

If the switch is 'on' (the knob is pulled out) this means the headlight circuitbreaker is overloading and flipping off. When it cools down, it will reset itself to 'on' and the lights come back on..... only to turn off again as the circuitbreaker heats up again.

Try a new circuit breaker, the old one may be defective. If the new breaker does the same thing, look for a short circuit somewhere in the headlight wiring.... if no luck, then try new headlights.

I'm having trouble finding the circuit breaker. Where should it be and what does it look like?

The breaker is part of the headlight switch on yours.

... and it should look like a small silver or grey metal box, no bigger than one-half inch across.

I replaced the headlight switch and it seems to have fixed the problem. Thanks! When I disconnected the headlight switch connector from the headlight switch I noticed the plastic plug-in and a few wires had melted a little but doesn't really look that bad from what I can see. Since it seems to be working fine now should I even worry about it or should the wires and plug be replaced?

I would look for a good used plug and connections your next trip to a scrapyard.