What is the true meaning of the word "roger" when used on the radio? What is its origin?
In the US, the "textbook" meaning is in the Pilot/Controller Glossary:


  ROGER − I have received all of your last  transmission. It should not be used to answer a  question requiring a yes or a no answer.


The AIM 4-2-3 gives an example of using it to acknowledge instructions:


  Acknowledge  with  your aircraft identification, either at the
  beginning or  at the end of your transmission, and one of the words 
  “Wilco,” “Roger,” “Affirmative,” “Negative,” or  other appropriate
  remarks; e.g., “PIPER TWO ONE  FOUR LIMA, ROGER.”


As a counterpoint to that, when I first learned to fly in South Africa I was told never to use Roger in transmissions because it doesn't tell the controller anything useful (obviously the FAA disagrees).

And as for the etymology, that's already been covered in another answer:


  This usage comes from the initial R of received: R was called Roger in
  the radio alphabets or spelling alphabets in use by the military at
  the time of the invention of the radio