Brian W Evans
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Here are two drawings that show the pinout, or the physical location of each pin, of the Arduino microcontroller and the most basic running circuit diagram. There are some tradeoffs by using this method, like having to program the chip in the Arduino before unplugging it and plugging it into the board but for minimal design this is a good option. (Although Paul Badger's Real Bare Bones Board is pretty small too and might save some work.)
The most complicated part of the process is wiring up the crystal or resonator. There are 2 possible choices here, both work pretty much the same but I find the crystal easier to hook up on a solderless breadboard or similar perfboard.
The rest of the hookup is pretty straightforward, just make sure the right pins are properly wired to either +5v or Ground.
There are a couple of components to make this work.
Crystal or Resonator:
If this is being soldered to a perfboard you need a socket for the chip:
You will not need a socket if this is only going into a solderless breadboard.
To power it up, you need a clean stable source of 5 volts DC. This means that you either pick up a switching 5V DC power supply from your favorite retailer, mail order one from Jameco or you will have to wire up a voltage regulator. Page 254 of Tom Igoe's Physical Computing has a nice layout of a 7805 voltage regulator with a couple decoupling capacitors.
For the voltage regulator you will need these parts:
Finally you will need something to stick all these parts into. Feel free to use anything you want: plastic, wood, a herring... The easiest method is to just use a spare breadboard, but the second easiest method is to permantly solder everything together using a copper clad perf board available at Radio Shack. These have holes drilled in a phenolic substrate with little copper rings, or pads, on one side.
Ok, that is basically it. I will update this as time permits with additional drawings and photos.