Why is part of this Derailleur loose? (Sun Tour Cyclone II) As you can see from this picture, part of my Derailleur  is loose.  Notice how the gear is not in flush with the cylinder. 

It has a no other parts (that I can find!) than a threaded screw to go in there, but the screw bottoms out or at least seems to before it pulls the assembly in tight.  

The screw itself is not of such diameter that it would hold the device in place, and while the geared part is secured by a metal lip on the side of the gear when you push it in place... the screw doesn't do the job.

This is my first time working Derailleur system.  I have no idea what part I'm missing.  I'd appreciate anybody who could tell me what it is I need..

Oh, and for the record.  This is the image of what the thing hooks onto.
It is hard to give you the exact reasons why this is happening, but your derailleur body lost a good mechanic connection with the cage. Maybe the screw that holds them has just became loose, or maybe it has gone bad, or maybe your derailleur is just old. Hard to tell from the pictures.
I don't want to be pessimistic, but repair will be hard, as manuals and parts are hard to find, and I suspect even if you successfully do it there will still be too much slack for this part to perform adequately.
It sure doesn't hurt to try. The only link I can point you to is for an overhaul of a Shimano derailleur, here. These components aren't too different and this should give you a help.
In the meanwhile, I would start putting some money on the side for a new one, the good news are that if you can't fix yours you can get something really cheap. If you ride a lot, it is better to invest in something a little bit better.
Good luck!