You may have many of the parts for this project already, but if you don't, the whole list costs ≈ $350, depending on where you get your parts from. Many, many of them can be salvaged, so look to recycle before you buy!
1 Arduino Uno or Diecimila
We'll be using this arduino to drive our stepper motors and servos. You can pick these up just about anywhere online. I got mine from Adafruit.
This is the most expensive item in the project. It'll be dealing with the inputs from each chess square to let the computer know where you've moved. We're using the mega here due to its speed and number of inputs. Adafruit
The mux shield (short for multiplexer) gives us even more
inputs for our arduino mega. We'll need 64 inputs in total, one for each square. Sparkfun
The motor shield will be controlling our stepper motors and servo. You'll need to solder it together. Adafruit
1 Large chess board with pieces
This one is a little more self explanatory. We want a large chess board here because the pieces need to be able to move in between each other with disrupting others. Make sure you measure the diameter of the bottoms of the pieces. We'll need that in a moment. I'm not sure where mine is from, but you can pick them up from a flea market for a bargain. The playable area of my board is 24".
Reed switches are magnetically activated switches. They'll help us find the location of moved pieces. NO stands for normally open, that is, the circuit is disconnected Digikey
16 10K 1/4 Watt Resistors
These are the pull up resistors for the built in digital pins. The mux shield, luckily, has integrated pull downs, so we don't need to worry about those. Digikey
Roughly 90 feet of 30AWG Wire
This is the hookup wire for all of our sensors. Radioshack
Neodymium Magnets to fit your pieces
This is where the measurements from the bottoms of your chess pieces come in handy. You'll need disc magnets to fit underneath each piece. For proper strength, they should be about 1/8" think. A great source for these is K&J Magnetics
. ≈ $55
This magnet will be attached to the XY table underneath the board, to move each piece around. K&J Magnetics $19
Note: This was Waaaay too powerful. It would draw in pieces it shouldn't have. You'd be better off going with some smaller ceramic magnets, like you'd find at Staples or another office supply store.
2 Pairs of 24" Drawer Bearings
The size of your bearings will depend on the playable area of your chess board. These allow for the stepper motors to move back and forth underneath the board. Amazon
Stepper motors can move in very precise increments. In the late 90s they were in just about every piece of tech you could find. The best place to get these are in old dot-matrix printers. You can them at the flea market for next to nothing!
The rack gears allow the stepper motors to travel on the drawer bearings. See the Step 4 for a more detailed explanation. Vex Store
This servo will be raising and lowering the powerful magnet below the board. You can find them at a hobby shop for ≈ $10, or Amazon
The perf board is super thin and will be the mounting surface for all of our reed switches. The price will vary greatly on this one, but I got mine from Home Depot for ≈ $5
1 2' x 2' x 1/2" MDF Board
Similar to the perf board, I got this from Home Depot for ≈ $5
Various lengths of scrap 1"x2" wood
This wood forms the bridge between the X-Axis drawer bearings. Go behind any hardware store and you'll see dumpsters full of this stuff for totally free!
This stuff is a godsend. It's used for just about everything in this project, from mounting motors to attaching the rack gears. I'm in love -- and I picked mine up from Radio Shack for $3
You probably already have this one, but if you don't, I picked mine up at Ace Hardware for $10 a couple of years ago.