This is a very easy, basic idea, however it leads to many more complicated codes and circuits.
Step 1: What You Will Need
3 x Cables
1 x 330 Ω Resistor
1 x Arduino board (it can be any kind, but for this instructable i used the UNO board).
1 x USB cable
1 x Computer with Arduino software installed (http://www.arduino.cc/en/Main/software if you haven't already got it)
1 x Breadboard
Step 2: Attaching the Power Cables and Resistor to the Breadboard
Then you need to plug one end of a wire into the ground ('GND') Power output, and the other end into the Negative section of the Breadboard.
Finally, you need to plug one end of the resistor from the Negative section of the Breadboard, to any horizontal line on the main section. (NOTE- It is very important to use AT LEAST a 330 ohm resistor with this circuit. If you don't, then you will almost certainly blow the LED.)
Step 3: Attach the Control Wire to the Breadboard
Step 4: Attaching Led
the way this is done is to put it so that each pin is next to a wire/resistor. This sounds unclear, but pictures should help
You have to put the LONGER pin next to the wire connected to the port, and the shorter pin next to the resistor.
Step 5: Plug Into Computer.
Step 6: Upload the Code to the Device
Step 7: And... Shazam
Note- if the Arduino is unplugged, then plugged in again, it will still work.
Hope you enjoyed the instructable, keep looking at my profile for more arduino instructables soon.