Q:

How do I hook up a new "pull-type" light switch in my basement utility room?

The old pull-switch was installed when we bought the house, and it was of a different type than the new one I bought.  I went to replace it, and found out I couldn't just swap the wires, since the old one had an extra connection.  The light switch box has this extra thing attached, and hooked in somehow, and I don't really know what it's for, or how to attach my new pull switch.  My new pull switch comes with instructions to attach the black (hot) wire to its brass screw, and the white (neutral) wire to the silver screw.  In the first picture, you can see the black wires are currently have a wire nut on, while the white wires are twisted around themselves, and exposed.  
Attaching the new pull switch would be straight forward except I don't have a clue what to do with the extra black wire coming from that other bronze box.
Here is an overview of the light box in the utility room in the basement:
You can see the bronze coloured, "extra" thing to the left of the box.  In the bottom right corner, in addition to the regular electrical wire leading away, there are two smaller wires that originate from this box.  I don't know where they go, since the wires are hidden once they go through the joist.
 
Here is a close-up showing the extra wire coming from the thing attached to the box:

Finally, here is a different view of that bronze box attached:
I'd appreciate any help someone can give (and any light you can shed on that attached bronze box).

A:

The extra brass colored box is a transformer for a low voltage system, probably a doorbell. One wire on the outside probably goes to the push button and the other to the chime. It could be some other low voltage device, doesn't really matter, it's definitely a transformer. The green wire appears to be a site installed modification to ensure proper grounding. The smaller black wires are for 120v power. One to house black, the other to house white. Which goes to which does not matter.
To install a lamp base with a pull switch, you need short lengths of white and black wires with solid copper conductors the same gauge as the house wiring. Connect the black one to the brass screw of the base and the other end to the other house black wires and one of the transformer blacks. Connect the white one to the silver screw and the other end to the white house wires with the other transformer black.
If the base has a green and/or hex shaped grounding screw, it is connected to the other grounding wires with either green insulated or bare copper wire. Since these sort of bases are usually non-metallic, there's usually no grounding screw. But if there is one, it needs to be connected.

