While many use a vacuum pump commonly known as a solder sucker for removing unwanted solder from joints, there is another method for achieving this that is far superior for some jobs. According to a retired British Army Officer, solder wick, or desoldering wick depending on where you come from, first entered use at the end of the second World War. Electrical technicians dipped the copper braid from coaxial cable in solder flux, then pressed the braid onto joints and heated them. The result was that the solder was wicked very effectively in between the fine fibres of copper. Rolls of woven copper braid pre-coated with rosin core flux are now sold specifically for desoldering use.
The stuff is fantastic, seriously. You just unwind enough to not burn your fingers, press it onto the joint, and heat. It's really quite hard to screw it up. Several points of note here though. As I said, unwind enough to not burn your fingers. It's copper, dude, copper. Not that if you overheat your wick, you will likely melt the plastic roll it comes on, even if you don't manage to melt through your fingers. Oh, you were holding it with pliers? Smart Alec, you still shouldn't be overheating the joint and risking damage to stuff. The other thing is don't unwind a ridiculous amount either. If you do, you are only unnecessarily oxidising wick. This is bad because oxidising copper reduces its conductivity. If you leave solder wick exposed to air, or more specifically oxygen, for a month or so, you can destroy it, rendering it useless. Treat your wick well, and it will serve you well. My current roll is about twelve years old.
As I said, solder wick is sometimes preferred over a solder sucker. A particular case where it's extremely useful is that you can place the wick across multiple joints, heat them, and sap the solder out of them simultaneously. Anyone who's ever attempted to do this with a solder sucker will know how painful it is.
Now go forth and desolder!