I'm trying to replace a ceiling light fixture. This house is 60 years old and the person who built it didn't always do things like everyone else.

There are 4 cables coming into the box in the ceiling, with 2 wires in each cable - black and brown (not white). The 4 black wires are tied together. Three of the brown wires are tied together, leaving 1 single brown wire. When I removed the old fixture, the connecting wires were the same color so I couldn't easily tell which was the hot, but both connections came off the brown wires, not the black.

On the switch that controls the light, there is a black wire and a white (not brown) wire. I don't know if that helps any.

So, what I need to know is, which of the brown wires is the hot one -- the group of 3 or the 1. Also, what would happen if I got it wrong and tried to turn on the light? Would it blow up or start a fire?

You said that the connections came off the brown wires not the black. By connections I assume that you mean wire nuts. I also assume that the bundle of black wires are undisturbed.
Do the wires at the switch appear to be the same type and age of the other wires, other than color? In other words do the wires at the switch appear to be original wiring?

I would say that the bundle of black wire are hot. Also that one of the black wires in that bundle goes to the switch. Therefore, the bundle of brown wires are the neutrals. The single brown wire is the return from the switch.

Based on that, you would connect the black hot of the light fixture to the single brown and the fixture white to the bundle of brown wires.

What will happen if I am wrong? Nothing bad. The light will still operate. The connection of the hot to the bulb will be through the screw shell rather than the center electrode. Not desired but nothing will blow up.

I would also say that the brown wires are not really brown but white. They have turned brown over time and due to heat in the box. The ceiling box is probably a small metal box with too many wires in it. I recommend replacing it with a larger box if at all possible.

That makes great sense! I will try it. And that's interesting about the color of the wire. It's all original wiring. Thanks a lot for the help!