The Arduino is a essentially a small computer that can fit in your hand. It's much slower and more low-power than a normal computer, but it does most of the same things:
  *  It can run software
  *  It can communicate with other computers
  *  It can control hardware - LEDs, storage devices, sensors, etc
  *  It can store data internally

Here is a picture of an Arduino UNO:

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There are 14 digital IO pins labelled 0-13. 
  *  Pins 0 and 1 are special and used for communicating with the computer via USB. You shouldn't use them unless you know what you are doing!
  *  Pin 13 is special because it is attached to an onboard LED. Try setting this pin to a high voltage and you'll see it light up!
  *  Pins 3,5,6,9,10,11 are special because they support PWM, a sort of analogue output technique.

On the left there are 6 Analog input pins labelled A0-A5
  *  They can measure 0V to 5V in increments of around 5mV
  *  They can also be used as digital IO if you really need some extra pins - their numbers as digital IO are 14+analogue pin number. So A3 is pin number 14+3 = 17