Things I use:-
- ANTEX TCS 50W soldering iron (because it has tips that have a large thermal mass)
- Good solder sucker/pump (and some spare nozzles)
- Desolder braid/Wick
- Desolder iron (like the one linked by Paul)
- Hot air gun (part of a soldering station)
- Electric paint stripper
- Snips
- Thin long noise pliers
- Long dentist type tongs
- Small flat blade screwdriver
- Two sets of helping hands
- Liquid Flux
How/what I use depends on circumstances. If it is a device with three or more pins/legs and I don't want it in usable condition I just cut all the pins/legs off and pull the remains out with the long noise pliers. Then clear the hole with the solder sucker. No damage to the PCB tracks
If I trying to remove a DIL chip and what to reuse it, I use either the hot air gun, or the electric paint stripper while holding the PCB vertically using either the holding hands or another method. After all the solder joints have melted, I gently pull the chip out using the long dentist type tongs. No damage to the legs of the chip, or the PCB tracks
For single sided boards that don't have plated through holes, just a iron & solder sucker are fine. After using the solder sucker, heat up the pin, then use the screwdriver to move the pin away from the tip of the iron so the pin is held in the middle of the hole in the pad. Then after it has cooled, use the screwdriver to check the pin moves freely is is not stuck with solder.
The worst things to desolder are connectors that use thermal plastic

and items that either are soldered to large ground or power planes, or which have large tags, or even worst, both
Mark