Think of the Arduino as a really smart switch. It has a series of inputs and outputs. The inputs can be as simple as a toggle switch or a push button or as complex as an optical sensor with discrimination to determine the light level in a room (in Arduino speak, a "shield"). The inputs can be used to tell the Arduino to "do something".
A quick example: 2 push buttons, an Arduino, and a servo. Push button 1 and the servo travels in one direction, push button 2 and the servo travels in the opposite direction. Push button 2 again and nothing happens. At the same time, the same Arduino can turn on an LED next to button 1 when button 1 is pushed and turn off the LED next to button 2. And do the reverse when button two is pushed. Or you could use 1 push button, add 2 LEDs, say a red and a green, and - push once, the servo moves in one direction and lights the red LED. Push the same button again, the servo moves in the opposite direction and lights the green LED.
Instead of controlling servos, the Arduino could control light bulbs or LEDs (say in a model of a bookstore) or streetlights or crossing signals, or, or, or ...
From the basics, as Olie said, its only limited by your imagination and the help you can find.