A hard drive spindle motor is a 3-phase AC motor, not a DC motor; if you can spin one of these up, in theory you could spin up any 3-phase AC motor (even the really large ones):
In order to turn one of these motors, you have to generate a phase sine-wave output, of the specific frequency for the motor, with each phase 120 degrees ahead (or behind) the other:
Most 3-phase stuff you see in everyday life runs off of 3-phase AC power, with each phase at a frequency of 50/60 Hz (depending on country); your home is usually a single-phase supply, but it is possible to run 3-phase stuff even then by using a particularly valued capacitor wired properly to give you the phases needed. A hard drive motor isn't much different, except for the frequency, which I would suspect to be much higher than 50/60 Hz (but I don't know what the frequency really is).
The four wires that come out of the motor are the wires that control the phases; one is a "common" wire connected to each phase (or stator coil; that is, the outer coil that doesn't move), while the other three are connected to "the other side" of each phase's coil. This configuration is called a "Y" configuration (I think it is sometimes also called a delta-configuration, but I might be wrong there), so called because the "center" of the "Y" is the common point, and the three arms of the "Y" each represent a phase coil load, with the ends being the other connections to the phase coils (and hence, a "Y" shape).
All you have to do to get started is to figure out which coil is which. The way to do this is simple: Set your multimeter to a low resistance reading level (ie, 200 ohms or whatever your low scale is), then place your multimeter probes each on one of the terminals. Note the value. Pick a probe as the "stationary probe" and note the pin number, then move the other probe to a different terminal. Note the pin number and value. Then move that same probe to the last terminal, a note the pin number and value.
If all three values are the same (or very close), then the lead you didn't move was on the "common" pin, with the other three pins being the phase coils (all resistance readings should be close to each other).
If you have a situation where one value is half as much as the other two values (approximately), then the pin where your stationary probe was is one of the phase coils end, and the position where it is half as much as the others the common point; the other two readings are the other two phase coils (double the resistance, etc - if you draw a diagram of the "Y" and put your probes on it to simulate what you are doing, you will see what I mean - at one point you only measure the resistance of one phase coil, while the other points you are measuring the resistance across two phase coils).
So - now that you know which are phases, and which is common, you can then take a look at this instructable:
If you notice half-way down the comments, someone posts a handy link to an Atmel Application Note on driving such motors; it is probably a handy document, so read it.
Something to keep in mind though to try, if you don't want to have to generate three sine-waves out of synch - is to use square waves; just kick a square-wave out at the proper time so that all are 120 degrees out of phase with each other, and the motor will probably spin (there might be a slight hum, it might not be as efficient or as powerful, but it will probably work!); hook up three of the output pins on the Arduino thru some transistors or MOSFETs (like you would a relay or DC motor), and the outputs of those to the motors phases (of course, your 12 volt power supply hooks up to the transistors/MOSFETs to supply the higher voltage/current to the motor's phases). Then write the code to generate the phases at the proper times for the frequency it needs (once again, I don't know what the frequency is for your motor; start at 20 Hz and move up, I guess).
Only one problem, though - you know which phase is which, but you don't know the order. But there are ways to guess; hopefully, you get lucky, and common is pin 1, and phases 1,2,3 are pins 2,3,4 of the motor - if it looks like that go with it. But if it doesn't seem to be set up that way, don't sweat it; carefully swap the phase wires around (leave the common where it is at), and note what the motor does when you run your code; you might try adding a potentiometer to the Arduino to allow you to change the frequency; spitting it out the serial port so you can take notes. Add a button you can press to signal the Arduino to switch the phases programatically (and spit out the order to the serial port).
At a certain point, with a certain frequency setting, you will see the motor twitch in one direction; keep playing with the frequency, and it should eventually spin. To make it spin the other direction, reverse the order of the phase pulses.
Hope this helps you!