To what degree does a student's undergraduate institution affect his or her chances of admission to graduate school?

If you graduated from one of the publicly known or any mediocre research universities,  (let's say top 100 national Universities or liberal arts colleges), you should be OK. I am an example to this.  I did not go to Ivy, Ivy equivalent or Big ten schools.  But I had several offers from top 5 and top 30 schools whereas my Ivy league buddy was struggling to get one.

As others have said, it depends on your field. My experience is in the social sciences, and here, the reputation of the program certainly can have an effect on the admissions decision. The reputation of a department, and particularly whether a program does discipline X like we do it (and think it should be done), matters a bit. The personal connections do too: if we have friends, former colleagues, or former students teaching in your program, we have a better idea of what kinds of things you have studied. The other side can be true as well, in a more general sense. If you didn't go to an R1 (research) university, or a solid liberal arts college, you're going to have a harder time demonstrating preparation to do research. But this isn't the most important factor. If you can demonstrate a dedication to research--including being named as an author or contributor to published research--this puts you in a much better light than test scores, grades, or your undergrad institution will.

I would say that needing to come from an elite institution during your undergrad is one of the biggest myths about getting into top graduate programs. At the end of the day, it doesn't really matter where you studied during your undergrad, rather what you did there. There are definitely benefits to studying at top programs, such as quality of professors, more research experience opportunities, and other opportunities that could make you a better candidate overall. However, the name on your CV isn't going to make much of a difference in the long-run. People get accepted into top Ph.D. programs from all sorts of different previous institutions. What matters more is your application as a whole rather than where you came from. http://www.theacademicadviser.com/six-myths-grad-school-applications/ Of course, transferring to a program like NYU is probably a sound decision, just keep in mind that what is more important is to pursue all the extra benefits you can have there rather than your previous university.

At Columbia, about 10 or 20 percent of students in my cohort seem to be from poorly-ranked undergrad programs. Admission committees love to see undergraduate research (2 or 3 papers if you can muster it). You should also have at most one non-A grade in your major classes.

It is not inherently about the "prestige" of the university. It depends on how well past students from your university have performed in their programs. With so many universities, marking, and evaluation schemes, it is nearly impossible to be entirely objective in the selection process. Quality of students (and by extension, the university) is therefore judged by how well past students from the same university have performed after selection. Under the same conditions of content and rigor that every other student faces, if your university student has performed well, it is likely that your university in general produces good students. This "prestige" is built over many batches. The benefit of doubt is therefore given to students from the more "prestigious" university.

It has huge benefits especially if you're an international student. Say, students coming from institutions which admission committee know, will have credibility for explaining bad grades or low test scores, no matter how rigorous the curriculum of some other student coming from a no-name institution is.

I received my bachelors from an undergraduate institution with a 90+% acceptance rate. I had competitive GRE scores, substantial undergraduate research experience, and a fairly sound application. So in my circumstance, my undergraduate institution was far and away the weakest component of my application. Yet, I still received multiple offers from great programs in my area of interest. So, at the very least, feel confident that one's undergraduate education isn't the most crucial factor with respect to getting into graduate school. Now, I should follow this up by admitting that, based on the scenario you posed, one's undergraduate institution would probably play a role, but again, I imagine it wouldn't be a critical factor. Why? The whole reason graduate schools use a multitude of measures one assessment by itself is entirely predictive of graduate school success. So, if in that scenario, two students are identical on every measure with the one exception being undergraduate institution, the graduate review committee may decide to bring both in for an interview, and thus, the interview would end up playing a more critical role in their decision process. If you are feeling uncertain about your graduate application process, I would recommend doing one or both of the following things. Contact people in the programs you're interested in and start a conversation. Also, find a paid/volunteer/internship/fellowship/etc.. that directly relates to what you'd like to do in graduate school. You can't control your undergraduate education at this point, but you can work to differentiate yourself in other areas of the application process.

I've done graduate admissions in a science department for many years (at a "prestigious" institution) and can say that the undergraduate department itself doesn't actually matter that much. We look for excellent students and excellent students can be found lots of places.  However, we do often rate letters of recommendation from people we know more highly than those from people we don't, and we are more likely to know people from big name places, so it matters in that way.  And often (but certainly not always) students have better opportunities to do research and other activities that we value at prestigious institutions, so that can help. So, if you can go to a great school as an undergrad, why not?  But if you can't, make the best of the opportunities you have.  Try to find summer research opportunities at different schools -- the more people who know you well and can write good, personalized letters the better. If you are at a smaller school, take advantage of the fact that faculty there are more likely to be able to get to know you better than those at larger schools and let them get to know you. Make yourself shine and you can get into good schools no matter where you are coming from.

I got into a good grad school possibly because my referring letters were from professors who worked with professors at the grad schools. Otherwise i has just average grades from a tough undergrad place (MIT). I dont think undergrad backgrounds mattered greatly in grad performance. People from lesser name schools did well if they worked hard and chose clever research topics.