You’ll need an Arduino microcontroller board. Several online vendors sell the Arduino, including Lady Ada and Sparkfun. The Arduino comes in several variations. The basic board is the Diecimila board, but there is also a mini and a Nano version, as well as the Lilypad. If you have never done this before, I’d recommend sticking with the lower-cost and simple Diecimila.
You’ll need a USB cable to connect the Arduino to your laptop computer. It’s a good idea to also buy a power adapter for the Arduino. You’ll find these items for sale at most of the vendors that sell the Arduino. You will also want a breadboard, some wire, a few LEDs, and some resistors. Lady Ada sells a “starter kit” if you’re willing to pay a little more for the convenience.