To best answer this question, it helps to understand the goal.
The main goal for obtaining a research degree is to help you land the best possible next position, either a job or degree, as well as your long term career development.
Both depend primarily upon your research ability as reflected on your publication record, and, to a lesser degree, the connection of your adviser.
Thus, you need to find an adviser who can best help you publish top papers and connect to top people in your fields.
For the former, look at your potential adviser’s publications. Are they of high quality and high impact? Never follow an adviser who does not have a very strong publication record, even in a top school. You will be much better off with a prestigious publication record from a less prestigious school than the other way around.
For the latter, look at your potential adviser’s collaborators, e.g. paper co-authors. Are they reputable researchers with diverse backgrounds (e.g. some working in academia while others in industry, and located in different geographies)?
For multiple advisers with similar qualifications, look at how strong their overall groups are. For example, if you want to do graphics, look at the graphics group of the potential adviser. Pick one that is stronger overall, e.g. with more and stronger faculties or publications.
Talk to the former and current students of the professors to understand their styles and personalities. If you cannot get along with your adviser, you are unlikely to be happy or productive. Judge the validity of the feedback though; for example, a weak student might unfairly complain that the adviser is too demanding.
Then, look at the geographic location of the school. You want to be close to the industry (job/internship opportunities) and have a good life while grinding away at your study.
The ranking or reputation of the department or school should be the last criterion, or when you do not have any idea what you want to do. In this case, go to Stanford if you can.
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