A brown box is one of the many various "boxes" known to the phreaking (phone hacking) community in the golden age of telephony, before the advent of modern phone line services. The name is a misnomer - it's actually just a method for doing 3-way and "party line" calling.
It's hard to imagine, but there was a time when there was only one phone per household. When that system was lifted, it left an ugly kludge still in effect: only one phone in the house could be in use at any given point. However, enterprising phreakers learned that if you crossed the wires of two phones in your house directly at the lineman's box, you could speak on both phones at the same time. You could even receive a call from one line, and then use the second line to call another number - in effect, turning your house into a phone relay.
Explaining the wiring of the first basic brown box is archaic now, but the basic process was simple. Each phone had 4 colored wires. By taking the red & green wires from one phone and patching them in to the terminals of the black & yellow wires of another phone would allow the patched phone to act as a passthrough phone. From here, the possibilities are endless.
This in itself is a variation of the classic beige box (wherein you took a phone and modified it so the red and green wires were loose, and then went to a lineman's box and phreaked in to make free phone calls) but done in the privacy of your home. Over the years, brown boxes became external and computer controlled, and relays were set up between houses (simulating call forwarding), which had the added of bonus of allowing a lot of phreakers to make long-distance calls at local rates through the magic of tracking your way to the right person.
Today modern household phones allow all lines to be active at once, and call forwarding and long-distance are practically de facto. It's hard to come up with a modern-day equivalent of the phreaker and his toys.