To remove components like this from motherboards is not easy, the biggest problem is lack of sufficient heat to melt the solder all the way through the hole, this is due to copper planes taking the heat away and acting as heatsinks, you need an iron with a much higher temperature than normal so you can remove the solder quickly and effectively without damaging the board.
As previously mentioned a good solder sucker works well provided you can satisfy the conditions above long enough that the solder is still molten all the way through the hole when you release the plunger.
You can get electric irons with suction which work well but are expensive and not viable for the one off or odd repair. The other method that usually works well for stuck pins is wire pins (you can buy tools with different steel pins), just melt the solder and push the pin through, again the solder needs to be molten and stay molten so you can do this.
Not an easy job any way, but the professional electric soldering irons with suction through the bit are easiest, the bits don't remain effective for very long though. Oh, and a tip, if you are doing caps and at least one hole is clear, then use the pin of the new cap to push through, cut the leg that is going through the clear hole shorter so that you can get the first pin through the blocked hole first, once the first pin is in the hole it should push through the hole easily whilst applying the iron to the solder side, you should clamp or have some one hold the board whilst doing this for safety - in case you slip and burn yourself.