Undergrad and Grad School at the Same University?
It's common for undergraduates preparing to move on to graduate school to face what may be an uncomfortable prospect—moving to a new university, with a different culture and unknown faculty and fellow students. Most successful undergraduates, although by no means all, become very comfortable at their undergraduate alma mater, and for good reason—they were successful there. But, if you're in that situation you'll often be advised that it's unwise to complete both graduate and undergraduate studies at the same university.
That was my experience. While I was at the University of Michigan, the three professors who had the greatest impact on me—Harrison Morton, Gary Fowler and Burton Barnes (*see note below)—all advised me to get experience at other universities (although that didn't stop Gary and Burt from trying to keep me at UM to be their grad student, and, indeed, Harry completed all his studies—B.S., M.S. and Ph.D.—at the University of Minnesota). And as a professor I've advised many undergraduates to go somewhere else for grad school. It's never because we don't want to keep them. In many cases, we'd love to have them stay. But it's not in the student's best interest.
Now, this is not "style over substance" nonsense. It's not that it looks better on your CV (**see note below) to have experience at different universities. It's that it is better.
Richard Feynman, the American physicist who shared the 1965 Nobel Prize in Physics with Sin-Itiro Tomonaga and Julian Schwinger (they were the main architects of quantum chromodynamics), faced the same issue when he was finishing his undergraduate work at MIT. He explained the situation very well in an interview with Jagdish Mehra, one of his principal biographers, published in Mehra's biography (1994; pg. 79):
I wanted to [stay at] MIT [for graduate school] because one learns at MIT that it's the best place in the world. I told [Professor] Slater [a professor there] that I wanted to go to MIT, and he said, "Why do you want to do that? You are not going to be accepted here." I asked him why. He said, "Because you should go to a different school for graduate work than for undergraduate work." He wouldn't let me in. And he asked, "Why do you want to go to MIT?" I said, "Because MIT is the best school in the country for science and engineering." He said, "Do you think so?" I said, "Yes." He said, "That's why you have to go to another school for your graduate work." Then I went to Princeton, and Slater was right. I learned that the world is bigger and there are many good places."
So, although you can go to the same university for both undergrad and grad school (if they let you), you don't want to.
Also, don't forget—when you apply to graduate school, you are applying to a person, not a program. Although it's becoming more and more common for graduate schools to have "programs" in which students do rotations among labs within a given academic unit (department, although it's now becoming politically popular to call these things something else, like "school") before they embark on their own research project, you still need to have a connection with a particular individual for your graduate work. So, never apply to a program that doesn't appear to have anyone there doing anything you find interesting.
* These three guys, Dr. Harrison Morton, Dr. Gary Fowler and Dr. Burton Barnes, were just fantastic and a real credit to the university. Harry taught forest pathology and was my formal adviser. He was the most pleasant guy and invariably helpful. Gary was probably the most important person in my undergraduate career. He taught biostatistics and forest mensuration (measurement). He helped guide my mathematical studies, as my course of study didn't require much math and I wanted more. Burt was one of my all-time favorite teachers. I've never met a more active, engaging instructor. Anyone who had the honor to take his forest ecology class will remember his "you gotta know the territory" opening lecture. All three were at the top of their fields—they were outstanding researchers, authors, teachers and mentors.
** CV is short for "Curriculum Vitae," which literally means "course of one's life." It's a detailed summary of your professional career, somewhat similar to a resume but much more detailed. Most people with advanced degrees (anything beyond a bachelor's degree) use a CV in lieu of a resume. If you are applying to an academic position or professional medical position, attach your CV, not a resume, to your application.
Reference
Mehra, J. 1994. The Beat of a Different Drum: The Life and Science of Richard Feynman Clarendon Press, Oxford, UK.