Solder suckers usually look like large syringes. There is a spring loaded plunger, and a button to release it. The plunger is pushed down. When you want to suck up the solder, you position the nozzle over the molten solder and hit the button. The plunger moves up, creating a vacuum and sucking up the solder.
Solder wick, on the other hand, has no moving parts.
It looks like wick used in oil lamps, except that it is made of copper. To use it, you put the wick over the joint and heat it. One thing to note about solder wick is that it is expensive, and because it is expendable, a solder sucker may be a better choice if you plan to do a lot of desoldering. I personally prefer to use a sucker to remove most of the solder, then finish up with the wick. Remember that when desoldering, the resin in the solder and the coating on the board may releases fumes.