The inverter is a very contrary logic Bit. It sends out the opposite of whatever it receives: send it an ON signal, and the inverter changes it to an OFF signal, or vice versa. Try putting it between two lights after a button: clicking will make the LEDs blink back and forth, like the lights on top of a police car!
The lights on top of a police car flash back and forth. A pulse sends a signal to both of the lights, but they flash opposite one another due to the inverter
Full lesson plan about logic available here.
Check out littleBits' Tips + Tricks for more ideas on using the inverter.