Just to answer some of the questions. . . .
The e-mail that asked us to write an essay for the Research Program stating our reasons for applying had 40 people on it. I think 24 people were given slots for the research.
As for what type of research, it varies but seems to be either basic sciences research (i.e. studying things on a micro level to find a link to oral cancer), epidemiological research (i.e. demographic risk factors for oral cancer), or clinical / applied research (i.e. testing certain biomaterials). The NYU website has an extensive research section with various categories.
I believe you can send an e-mail to the director of the research program and enquire about getting on the waiting list but don't take my word for it.
Hope this helps,
Bob