The Arduino is a microcontroller, an integrated circuit to control electronic devices that you can program yourself. The Arduino consists of hardware combined with an open-source computer platform. The Arduino can be connected to analogue and digital input and output. You programme the microcontroller controller in a C++ variant in a programming environment (Arduino IDE). Then you compile the code to machine code and upload that code to the Arduino to have it executed there.
Peter shows you how he uses the Arduino IDE on his Linux computer, programs a piece of example code, compiles it and uploads it to the Arduino to be executed. Using an Arduino Uno (Rev 3 is the current version), laptop + USB cable to connect the Arduino, a breadboard with some wires, LEDs and buttons.