Several people in One Piece have the middle initial of "D"
The things they have in common are their fearlessness, the will to do what they want to do. But Whitebeard said that Marshall D. Teach is not a real "D." because he is scared to death, and possesses evil within.

The Story

The great kingdom possesses vast knowledge and prosperity, this make the other nations afraid of their power, and so they try to destroy it. The kingdom was not destroyed easily. Because there was a group who fought against the world government to protect the kingdom. And the proof is in the Poneglyphs. That's why the World Government is afraid of getting anyone involved in studying the Poneglyphs, That's why the study of Poneglyphs are forbidden.

So what is the connection of the "D" here? they are the descendants of the group of people who fought against the world government.