Your advisor and relationship with this person will decide a great deal of your success and happiness in graduate school. Therefore, it really should be your top consideration when deciding upon a school. If you can afford it, go visit your top 2-3 choices of schools, and meet with a potential advisor and some of their graduate students. Also, explore the town or city.
A great advisor will be a mentor in all senses of the word. They should regard you as a colleague in training and be willing to guide you intellectually and sometimes personally through graduate school and intensive research. This includes helping to find you funding, co-authoring research papers, and being an advocate in your dissertation research and write-up. A good advisor will pass on their intellectual tradition of how to successfully develop and conduct research, and how to fund that research as well. Look for an advisor who has reached tenure, as this will likely ensure that they will not leave in the middle of your research and generally means they are more established in terms of lab space, money, and respect. On the flip side, make sure the advisor will not retire during your studies.
Bad advisors are generally indifferent to you and your concerns. This usually adds years to the length of your study and may cause you to become frustrated and disillusioned. A horrible advisor may even take credit for your work and harm your reputation. You should not be afraid of or unduly uncomfortable around this person. Listen to your instincts, and if not your own, then those of the advisor’s current graduate students.