She explained ghosts and the veil, but you had to scrutinize.
At king's cross, it was a representation of a Purgatory if you will. I truly believe that Dumbledore had actually waited there for Harry, seeing as how he was not "dead" in the sense that he still had his Hogwarts painting to communicate to people with. The story from the Gray lady is also integral to this. I believe that when someone who has a very strong tie to the earth dies and doesn't wish to past on, their spirit CAN become a ghost. However, that's not the only prerequisite, as I believe remorse plays a very serious role (i.e., the baron killing the Gray lady, the gray lady feeling remorseful about her mother, Nick feeling remorse for not only not dying like a real man, but also not being able to join the headless hunt), but this is just a theory.