Why apply to private universities?
For top students, private univerisites can offer better financial aid than local state schools. Private schools can provide better opportunities for undergraduate research in the area that the student is studying in. The professor to student ratio tends to be smaller than that of a state school, and relationships with those professors can yield a greater chance of being involved in projects that will help build a greater understanding of the subjects being studied.
Where do I find more information on these schools?
Common Application - https://www.commonapp.org/explore-colleges
Big Future - https://bigfuture.collegeboard.org/college-search