Judging the by the variety of YouTube videos on the topic, guitars must lose their headstocks fairly often. (The headstock, by the way, is just above the neck (the part where the frets live), and is where the tuning machines live.) The grain of the wood runs the long way along the next, naturally, but the headstock angles back toward the player, so the strings are pulled toward the neck (through the "nut", which has grooves to make sure the strings stay where they're supposed to). Thus, there's a joint where the angled wood of the headstock meets the neck, and it tends to be fragile. (Imagine a guitar lying on a bench. Where does it break when someone sits on it?)

The break consisted of a long, smooth section where the wood at the top of the next separated along the grain, and a jagged section just below the first set of tuning peg holes where the wood broke across the grain. I used a sharp chisel to carve away some of the jagged bits, which would never mesh with the original wood, enough to match the large smooth areas together again. Then I prepared clamps, coated both sides of the joint with Titebond carpenter's glue (indoor formula), fitted it together, and put on the clamps. As glue squeezed out, I wiggled the pieces into nearly perfect alignment, wiping off the excess glue as it appeared.

Then, I set the assembly aside while I tidied up the work area, not realizing that I'd made a big mistake. A few minutes later, I checked on the joint to wipe off more glue, and discovered that the wood had slipped along the joint. With hindsight, it made perfect sense: a diagonal joint, squeezed on parallel sides, overcame the friction of the clamps to slide along the diagonal.

Fortunately, the glue had not yet set, so I was able to loosen the clamps and slide the joint back into position. Then, I lashed up a length-wise "bar clamp" longer than the whole instrument to counteract the slippage, re-attached the original clamps (maybe not so tightly), and let the glue cure for 24 hours.

After that, I sanded down the rough spots (excess glue and wood fibers) at the glue joint, cautiously reattached the strings, and tuned it up for playing. Once in tune, I let it rest for another day or two to make sure that the string tension (typically around 150 lbs. total) wasn't going to pull the joint apart. It held the tuning.

Finally, I strolled back down to the house where I had found the guitar, and knocked on the door. "I'd like to speak to the person who put a broken guitar out into the trash last week. I have repaired it. Would they be interested in buying it from me?" A brief negotiation ensued. "How much will you pay?" "How much do you want?" "$50?" "$30?" "Good enough. I'll go back home and get it." When I returned with the instrument, the owner gave me $50 after all, having had time to think it all over, I guess.

So, I'm pretty happy about all this. A nice instrument was salvaged from the trash. I got a chance to refine my repair skills. My neighbor (Asian, probably a grad student) got his guitar back. I had an excuse to visit the local guitar shop to get replacement strings (mostly because I forgot where I left the originals when I took them off.) I had an opportunity to show generosity to my neighbor, and he in turn was generous to me. I got a story to tell, and now I've told it.