We start to see the meaning behind the D. In the episode 703 of the Anime...
...Corazon talks to Trafalgar D. Law, telling him how young kids in Marie Jois are instructed to see D. as a sort of boogeyman:
"Badly behaving children will be eaten by the D."
He continues saying that in some parts of the world, the Family of D. is known as 神の天敵 (kami no ten teki), translated "God's natural enemy".
He also states that "God" in this instance is used to refer to the Celestial Dragons.
It hasn't been revealed yet I think.
The initial was first pointed out in Oda's SBS questions and answers corner, he was also asked what the D in Luffy's name stood for. Oda just replied to read it as a D for now and that he would reveal the truth behind it in time.
You can read more in the Will of the D. wiki.