czechychicky: How important is a PhD supervisor vs. university at which the degree was obtained?
chemistress: How much does the graduate school you attend matter? Will I seriously be hurting my future job prospects if I receive my PhD from a (much) lower-ranked school?The old adage is that your terminal degree/experience should be the best one possible. There's a lot of caveats there, but I think that's it is broadly true.
If you want a classic Ph.D. chemistry job, the conventional wisdom is that your institution matters a lot. For Ph.D.-level jobs in the US chemical/pharmaceutical industry, it might even be true that the Ph.D. supervisor matters more than the institution itself.* I don't know about other fields in chemistry; it's possible that they are not nearly as exclusive. (I should hope not.)
Here's how I look at the difference between an adviser and the institution when it comes to jobs: your institution will help your CV (the "prestige" factor, networking, potential internships with nearby employers, other professors to be your references, facilities), while the interaction between you and your adviser is what makes your thesis and makes your job talk. Which is more important in getting your job? I would hope it's your science and your skills, but let's not be too naive. It's probably best to maximize both.
Will you be hurting your job prospects if you get your Ph.D. from a lower-ranked** school? I don't really know, but there is probably a wide gulf in the distribution of job offers and salaries between graduates of, say, the top 5 or 10 institutions (the Harvards and the Caltechs of the world) and everyone else. Is there a big difference in salaries and lifetime earnings between the 50th ranked school and the 100th ranked school? Possibly, but again, I don't really know. It is also quite possible that if you get a Ph.D. from a lower-ranked school, you're probably increasing your chances that you might need a postdoctoral stay at a higher-ranked school to polish your CV.
[If you haven't noticed, I keep saying "I don't know, because there's no good data." This is basically the story of the paths in our field. While individual groups/professors can be quite good at keeping track of their alumni, neither universities, employers or employees (or the American Chemical Society, for that matter) seem to be interested in gathering, synthesizing or divulging data in an attempt to inform younger people. Why that might be is left as an exercise to the reader.]
There's something that I haven't mentioned yet, which is success. It's a really difficult thing to define, but it will probably mean 1) publishing a lot of good, well-regarded science and 2) learning to be a high-quality, independent scientist. I think that's hard to do anywhere, but some institutions and professors fit you and your personality more than others. So that's going to be my spin on the above adage: you should go to the best school that will help you be the most successful.
*UPDATE: Experienced commenters disagree with me and seem to agree that in industry, the school matters more than the adviser name.
**Whose rankings are we going to use, for that matter?