The three most common inexpensive ways to remove solder include a “solder sucker”, solder wick, and an iron with an attached desoldering bulb.
Solder Sucker: To use, press in the plunger, heat up the joint, and as quickly as possible, pull out the iron, place the sucker’s nozzle over the joint and press the release button. It may take a few iterations to clean out a joint enough to wiggle free a component. On plated through holes it may actually be easier to add moresolder to the joint before sucking, since the sucker won’t be able to create a vacuum with too little solder remaining.
Desoldering Iron:This works just like the solder sucker, but is much easier to use because there’s no race to suck out the solder before it freezes, as with the “solder sucker”.
Solder Wick (desoldering braid):The above two tools will often not be able to completely remove all the solder. Many people use the sucker devices above to remove most of the solder, and then get the rest with solder wick, which absorbs solder through capillary action. It’s a fine weave of copper wires that are coated in flux. Place it over a joint and then heat from the opposite side with a soldering iron. It may help to have a bit of solder already on the iron’s tip to speed heat transfer.
Surface mount desoldering: Surface mount chips are especially hard to desolder because it’s very difficult to completely remove all the solder pin by pin, and avoid overheating the board and lifting a pad. Professional shops use expensive hot air guns or special tips (shown below) to heat all the joints at once.
There are fortunately a few cheap ways to desolder surface mount chips.
ChipQuik provides an interesting solder that when melted over existing joints produces a new low-melting point alloy (under 200 °F) with a much longer solidification time. The longer solidification time enables you to melt all the joints at once and then flick off the chip.
There are many guides for making a DIY hot air gun with the RadioShack® desoldering iron and a fish tank air pump. Engadget has one of the better how-to guides. Here and here are some more plus the supposed original.