I have one of these USB hubs with several USB connectors, which plugs
into a single USB port on a computer. It dont work. If I plug in a flash
drive directly to the computer, I can access that drive. But if I plug
it into the hub, the drive never shows up. I tried this on 3 computers.

I opened that hub, and there is a circuit board, with each output port
connected to a trace on the board which goes to a black dot on the
board. I dont know what that "dot" is, but I assume it's a chip of some
sort, and it looks like the dot itself was some sort of epoxy placed in
the board.

This was a gift, and it's going to get returned to the store, but I am
just wondering what kind of circuit this is?

Until looking at the innards, I thought that these hubs were just a
direct connection to the input (from computer) USB. But I see there is
more to it....