A couple of methods I've used over the years when doing repairs and making my own projects:
1) A good quality (i.e. aluminum body) solder sucker. Just remember to clean it fairly often. Great for cleaning out holes after you've removed a part.
2) Solder wick. This sees very occasional use when nothing else works (i.e. removing solder bridges between pins of a surface mount IC). You can make you own by stripping the ends of any stranded wire and dipping it in soldering flux. Not as fancy, but works just as well.
3) Heat the solder you want to remove to until it melts, and then immediately give the item you're removing it from a light tap on the table (or in some cases a slightly harder tap). I've used this method also for cleaning out holes in circuit boards. Noisy but effective.
4) Desoldering through-hole (DIP) ICs-cut all the pins off the main body of the chip with a small cutting pliers. Remove the chip. Take the pins out of the holes one by one by heating each pin in turn, and yanking it out with a tweezer. When you're done clean the solder out of the holes with a solder sucker, clean the board with alcohol, and solder in the new IC. I've replaced 40-pin chips in under 3 minutes using this method.
5) Removing surface mount ICs-apply a liberal blob of solder to both sides of the chip, bridging all the pins. Jump the soldering iron back and forth between the two sides. The goal is to have the solder blobs on both sides molten at the same time. When this happens, quickly grab the IC with a tweezer and yank it away from the board. If all the solder is molten it should come right off. Clean all the excess solder from the pads using a solder sucker or solder wick. Install the new IC, being careful not to bridge the pins. If you do bridge the pins, clean it with solder wick. This method works great on most surface mount ICs up to maybe 20 pins. After that they're simply too large to keep the solder blobs molten on both sides for yanking off. You need special SMD desoldering equipment. You might need a magnifier to see what you're doing. I can solder 50 pin per inch SMD ICs with my naked eye, but that's just me. I know people who have trouble with DIP chips without a magnifier. Oh, and lay off the coffee for 24 hours if you'll be doing this. Hand tremors can cause havoc when dealing with stuff this small.
6) Removing most two or three leg components-heat both pads simultaneously, yank component out, clean holes, put in new component. Similar to (5) except fewer pins.