Adding on to abhiarora's answer, if there are no lights at all on the Arduino, and you verify your cable is outputting power, then you may have shorted the device and need to replace the USB port diode. On my Arduino Nano, this diode is directly under the USB-mini port, on the bottom of the board.
Picture of diode:
I replaced this on my Arduino Nano with the CDBW0520L-HF from Comchip Technology and the Arduino worked perfectly. (Link provided is from www.arrow.com).
And in case you have a Chinese Arduino Nano like me: