loose brake calipers
As Daniel R Hicks pointed out; that style of brake caliper can be a pain to work with, however there are a few things that you can do:

Make sure that the calipers can rotate side to side, they're supposed to be able to do this. 


This is intended to work as a self centering system.
Imagine the brake being set up off center(where one pad is much closer to the rim than the other). When you hit the brake one pad is going to the hit the rim first. In the case of the brake pivoting around the mounting bolt the other pad will be forced toward the rim. If the brake is fixed at the mounting bolt, then the frame will resist the force coming from the pad/rim interface, and you will essentially be braking with only one pad.
The mounting pivot usually consists of an acorn nut and locknut system. The locknut sets the appropriate pressure on the brake arms to eliminate play while allowing the system to loose enough to pivot smoothly.


Make sure all of the pivot points are smooth


Clean out the pivot points as best as you can and apply a little oil(chain lube works fine)
I wouldn't take the calipers apart, even if you know what you're doing, it's a pain and usually not worth it.


Finally, you can re-bend the springs to center the brake


This is best done with the Park Tool OBW-3 
Bend the spring when the brake is mounted. Make small adjustments, and keep re-checking to see if the brake is centered. You want fatigue the spring as little as possible. 
Be careful doing this! Old springs are usually oxidized and you can snap them when you re bend them. It is also possible to snap the washer that clamps down on the spring.