So - the iron I talked about was the soldering iron.
I use one of these:
These are most popular:
You could un-solder, but I personally wouldn't worry about that.
Here is a solder sucker being used (the green thing in the person's right hand):
and here is a de-soldering wick:
Sometimes a solder sucker looks more like an infant snot sucker, but I've never used that kind.
SO - how to fix...
After cleaning, heat your iron up. Apply a dab of flux to the area of the board you are going to be working with - a small dab on each pin, for example:
I apply flux with a paper clip or toothpick.
Take your hot iron, and place it against the first pin. Hold it there, and the flux will melt and thin out, and then bubble - and then the solder will begin to melt. Don't hold your face over the smoke that comes up... I've never been harmed (that I know of!) by it, but it isn't the most pleasant to inhale.
Watch the solder get hot and liquefy, it will get shiny, then remove the soldering iron from the pin. You don't want to overheat it (melt the plastic that forms the socket), but you don't want the joint to be too cold either. As the solder cools, it will frost over in appearance. Do not shake or wiggle the board while it cools. When the first pin is cool (won't take but a few seconds), move to the next pin.
When you are all done, wash the board again with rubbing alcohol and check your work.
Under magnification, you may be able to see the cracked solder joint.