As mentioned earlier, make sure there is sufficient solder on the joint before desoldering. This is especially critical with a de-soldering gun/station.
If you did not add solder to the joint, clean the joint with a fiberglass pen and add some flux to it. Select a nozzle that is the same size as the pad and then preheat, clean and tin your nozzle. Clean the nozzle of excess solder frequently while working.
Surprisingly I see almost as much damage done by people using a desoldering gun as those using other techniques. This is typically caused by insufficient solder on the joint, which leads to pressing the nozzle directly in contact with the land in an attempt to get good heat transfer, or otherwise placing the tip in contact with the pad. The Hakko manual specifically cautions users about this (see second image below).
Using a desoldering gun does not guarantee success. Even with the Hakko if you pry the chip up while there is still contact adhesion you are going to damage a pad.
If you have trouble maintaining a gap, slide the end of a toothpick under the nozzle as the solder melts. As you get more experienced you won't need this step. If the pitch of the vacuum changes while desoldering, you are likely pressing the nozzle against the board, which restricts air flow and causes the change in sound.
Just as with soldering, the most important thing is to create a solder bridge (in this case from the melting solder) which transfers heat more effectively than nozzle to land contact. If the joint is low on solder you will not get a good bridge and should add some solder to the joint before desoldering. It is even OK to add solder to the point where the joint would normally be considered to have too much solder. In fact if you're having difficulty desoldering a component you may find that helps with the process.
Place the nozzle on the joint and once the solder has begins to melt (1-2 seconds), gently swing the nozzle back and forth. Once the lead moves easily, indicating the solder is liquid, press the trigger for 2-3 seconds. On more difficult joints (large traces like ground and Vcc) I may give it a couple of 2 second hits. Remove the nozzle from the joint and continue to press the trigger for a couple of seconds to clear the nozzle.
Total time on the joint for small components should be around 5 seconds.
The through-hole should be completely free of solder when you're done. Turn the board over and make sure the pad on the component side is also completely free of solder.
In the image below you can see where the right and left leads of the IC still have some solder. If this occurs, repeat the procedure above, but keep in mind you may need to add some solder first. As with the solder sucker, the desoldering gun requires a certain amount of mass to work efficiently.
If this occurs frequently then you're not getting a good solder bridge (too little solder on joint), the gun temperature is too low or the joint is not being heated long enough before turning on the pump.
Once the hole is visibly clear take your orangewood stick and press each lead on the non-component side of the board away from the side of the pad it is resting against. This will break any contact adhesion. Repeat on the component side of the board. If you are having trouble with a lead you can heat the joint with a soldering iron while pushing the lead towards the middle of the hole with the stick. The latter approach is also safer, especially on a high-risk board.
Once you get better at desoldering you can try grabbing the chip with a pair of pliers and rotate it slightly clockwise and counter-clockwise to break the contact adhesion. Not recommended until your skills have improved.
Once you are positive the leads are all free, gently pry the chip up. Clean up the flux and clean the desoldering gun as described earlier. During long desoldering sessions you may have to perform this cleaning multiple times. If the gun is not cleaned and maintained properly it will dramatically affect performance.
One tool the professionals use to remove contact adhesion on chips they know are troublesome is a hot air rework station or a hot air gun. After removing the solder with a desoldering gun they will use a hot air gun to heat the joints until they can easily remove the chip by pulling on it. You can pick up an Aoyue 8032 hot air gun for about $80. I normally would not recommend Aoyue, but a hot air gun does not require the same quality of design and manufacture as other soldering equipment.