When looking for resistors, what is the `W` for?
Yes, the "W" is for Watts.  The Wattage rating represents the total power (in Watts) that the resistor can safely handle without overheating and destroying itself.

You can determine the needed "W" rating by calculating the total current flow through the circuit, at a given voltage.  Wattage is given by voltage (in Volts) times current (in Amperes).  Use Ohm's law and your knowledge of the supplied voltage and any other resistance in the circuit to calculate the current, and then converting that into Watts is straightforward.  Just watch unit conversions... eg, if you're working in milliamps or millivolts, do the necessary conversion(s) as part of your calculations.

This site has more information on the topic, that you might find useful:

http://www.csgnetwork.com/ohmslaw2.html

For a number of introductory electronic projects (the kind of things you'll find in online tutorials and intro books), the 1/4W resistors are often sufficient.  Also note that you can increase the power handling capacity in a circuit by using a matrix of identical resistors.  Or you can just buy resistors with a higher power rating.