Solder wick, coffee, and paper towels keep SparkFun running. You can steal someone's diagonal cutters for a minute, but you'd better ask before touching their solder wick.
Solder wick is a composed of copper threads braided together. Flux is normally added to help the solder flow from where it is not supposed to be (the jumper) to the wick. Hence, solder wick is sometimes called desolder braid.
Yes, you really can solder anything, and solder wick is a large part of the process. TechSpray makes a very good product. This is what we use in SparkFun production. We've tried the cheaper wick and it just doesn't work.
This wick is 0.055" (size #2) wide and 25ft long. As you use it, the bits of wick can't be reused but 25ft should last you a year or more.
Be sure to checkout the SMD Soldering Tutorials to see how to use solder wick.