Assembled Arduino board, preferrably an Uno or Duemilanove (or whatever the latest version is)
Arduino compatibles will work but there's a lot of issues with ultra low cost 'Arduino compatibles' (e.g. eBay, Amazon, etc) where they have shoddy substitutions that can bite you later. It's good to have at least one known-genuine Arduino UNO!
You can also use an Adafruit Metro which is a drop-in replacement for the UNO, some components like the LEDs are in different locations.
USB Cable, any length. The cable should match your Arduino's USB connector. Official Arduino UNOs use USB "Printer Cable", a blocky cable. Some compatibles use USB Mini-B or Micro-B.
A plug in wall adapter is handy if you want to run your Arduino project 'standalone'
You can use any adapter that is Center Positive and 7 to 12VDC output - we recommend 9V DC if possible Check the second photo for the symbol that indicates Center Positive and for the 9V output text. The Arduino is fairly rugged and can survive plugging in the wrong adapter as long as the voltage isn't higher than 20V but it's 'stressful' to the 'duino and you should aim for 9V!
4 rubber bumpers
OK these are really optional but we include them because they're so nice to have on an Arduino and they help keep your table from getting scratched up. You can pick these up from a hardware store