kipp:
http://arduino.cc/en/Tutorial/AnalogInput
When I use a 10k or a 5k, it still perfectly makes the serial output from 0 to 1023. I would think it would get to 1023 only moving the wiper HALF way with a 5k, but it still takes a WHOLE spin of the shaft for it to use the full range of numbers (0,1023). How does the Arduino doing this?
A potentiometer is a resistor with a movable "pick-off" point.
If you turn the shaft 1/2 way, it's "picking off" approximately 1/2 of the voltage applied to it. Since 5 volts is applied to "all of it", taking off 1/2 gives you 1/2 the total, or 2.5 volts. Looked at from this angle, the value is not important.
However, the value of the resistance DOES matter to a certain extent. For example, if you used a 10 ohm potentiometer connected across 5 volts, the pot alone would draw 500 milliamperes and dissipate 5 * 0.5 or 2.5 watts (i.e. it would get quite warm and maybe even burn out).
So, too LOW a value is not good (although if you turned the shaft 1/2 way, you would STILL get 1/2 of the total 5 volts out - if the pot wasn't burned out that is).
On the other hand, you could use a 10 megohm pot. Again, if the shaft was turned 1/2 way, you would get 1/2 the output. But, the input impedance (or "resistance") of the A/D input pin itself is less than infinity. In fact, Atmel recommends that the impedance of the signal you are measuring be 10K ohms or less.
With a 10 megohm pot, setting the shaft at the 1/2 way point would still give you 1/2 of the input voltage, but if you connected that to an Arduino A/D input pin, the A/D input would "load down" the signal and you would read a lot less than the 2.5 volts (or an analog count of 512) that you expected.
So you see choosing the RIGHT value of potentiometer involves a compromise between having a "stiff enough" (low enough impedance) signal to measure accurately while not being too low and drawing a lot of current and possibly overheating.
For use with the Arduino A/D inputs, a pot value in the range of 1K, 5K or 10K will give you the best results.
Make sense?