The Raspberry Pi is an amazing 35 dollars mini-computer. It allows you to do everything you could do with a regular Linux computer (Connecting to the internet, watching videos, launching applications, ...) but also to interact with the world surrounding it, just like an Arduino. That's why I qualify it as a mix between a computer and a micro-controller.
That's also why I chose it for this project. I'm going to show you how to control LEDs with your Raspberry Pi. Firstly directly from the Raspberry Pi itself, then from any device in your house like your Smartphone or your tablet.
Here is a very good example of what you can achieve after reading this Instructable: http://www.instructables.com/id/Web-Controlled-8-Channel-Powerstrip/. Thanks to Rleddington for his amazing project.
Update: French version here
Step 1: The electronic part
Concerning the resistors, they should be 270Ω but since I don't have this precise value, I'm using 560Ω resistors (LEDs are just less bright).
Finally, I've made two schematics to make it simpler. The first one (with the complete Raspberry Pi) is showing you the real pins as they are shown on the board. The second one is a simplified version, it's showing you only the useful pins and their matches in the Wiring Pi library (GPIO Wiring number/Actual number on the board).