Neighbor's Airport Extreme extending my wireless network?
After the neighbor and I spent some time going over our networks, we figured out the problem: we were both extending our home networks over the house power lines, and apparently the street power for our row houses doesn't provide much separation.

The HomePlug system, which is what both of our adapters conform to, allows you to set up encryption on the network traffic.  Not only does this keep things secure, it also keeps other networks from getting tangled.  However, both of us had gotten systems that default to no encryption and we'd blithely left them that way.  (Mine is a Netgear Powerline AV 500, my neighbor has some other manufacturer's)

As Thorbjørn suggested, the network graph that the Airport utility was showing was just a source of confusion.  Once we'd separated the HomePlug devices, everything went back to normal.