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Also known as a 'solder sucker' or 'solderpault', the solder pump is a device for removing solder from a through-hole solder joint1. Depressing the spring-loaded plunger locks it in place. When the button is pressed, the plunger releases, sucking solder up through the nozzle. Inside the body of the solder pump, the solder disperses into tiny droplets, which immediately freeze into a powder. Depressing the plunger past the locking point drives a metal spike through the center of the nozzle, cleaning out any solder that may be stuck there.
The nozzle of a solder pump is usually made of a cheap thermoplastic, and considered a consumable. Far better to melt a cheap nozzle than an expensive component. In high-end anti-ESD solder pumps, the nozzle is made from electrically conductive teflon; when held by an operator using a wrist strap, any static charge on the nozzle is quickly dissipated to ground.
To desolder a component using a solder pump, hold a soldering iron in your dominant hand, and the solder pump in your other hand. Depress the plunger with your thumb, and hold the nozzle ready over the solder join. Heat the solder join with the iron, and when it is liquid, quickly withdraw the iron, press the nozzle of the solder pump to the join, ensuring a good seal, and press the button. If the seal was good, all of the solder from the join will be sucked up by the solder pump. Move the pump away from the work before depressing the plunger again, in case solder powder falls out the nozzle.
If the seal was bad, and solder remains, re-do the joint with the iron and solder, and then try the removal again. Solder pumps depend on surface tension to remove solder in the through-hole of the board, and may not remove it all if there is not much solder on the surface. The board can be damaged by pulling out a component that is still soldered to the through-hole.
1 - Solder pumps can also be used for solder bucket joints, but it is Not A Good Idea to use it on surface-mount components - there is a good chance that the tiny surface-mount component will be damaged by the heat of the iron, or even sucked into the body of the solder pump! | 
