I need to solder some headers onto an Arduino shield - what's the recommended wattage for working on that scale? I ask because the shield is already populated with ICs and I don't want to burn them out.
Selecting an iron that is too low in wattage requires the user to apply heat for a longer amount of time. This actually makes the surrounding area heat up more than if a more powerful iron were used to heat the component. Although a 25W iron will likely do this job without any trouble, and 80W temperature controlled iron is much more versatile. Make sure you use flux, and tin the iron afterwards.
The correct choice is a sufficiently powerful temperature-controlled iron (50W, say) set to the right temperature for the solder used, with an appropriate size tip. I use Metcal equipment and they recommend a tip roughly the same size as the item being soldered, for optimum heat transfer.
You generally want the biggest iron that still lets you work on the parts easily. You then want to put the biggest tip on the iron that still lets you work on the parts easily.
Overall power is not important, really. The critical factor in a soldering iron is the instantaneous heat capacity, e.g. how rapidly it heats up what it is in contact with.
Basically, a slightly too hot iron is better than a slightly too cold iron. Basically, if you can make each solder joint quickly, less heat is conducted up the leads of whatever component you are soldering, to the IC die.
Basically, there is a non-trivial amount of thermal resistance between the IC pins and the die. Therefore, the pins spending 10 seconds at 600F is worse than 5 seconds at 750F, since the internal temperature effectively follows the same curve as a RC filter.
To add to the other answers, I would be cautious if you're soldering on a board where you have an internal power/ground plane or some other sea of copper. Remember that the more copper your pin comes into contact with, the more heat dissipation will occur and the longer you need to heat the pin before the solder will flow properly. I haven't creamed a chip this way but I have noticed my board makes a nice temporary hotplate after soldering around copper planes. Conversely this may bite you if you then turn to pins hooked up to normal traces and use the same technique to solder those pins in that you did with the power/ground plane pins.
Also when soldering fine pitch parts (like a QFN package) I find solder-wick invaluable. That way I can allow some amount of bridging between pins and 'soak-up' the excess with the solder-wick. Usually I find the solder-wick will definitely soak up the bridge and leave just enough to make a connection between pin and pad which I either leave or touch up with a little additional solder.