What is the polite way to notify a graduate school that I do not intend to enroll?

I had to do this. As I recall, I wrote something like this:

Dear Whoever:

Thank you very much for your letter offering me admission to the PhD program in Whatever at Fabulous University. While I am grateful that the committee responded favorably to my application, I have received other offers. After much thought, I have decided that a program at another school will best address my interests.

Again, I very much appreciate your consideration and your offer. The decision was difficult, and I want to assure you that I do think very highly of the program at Fabulous University.

Sincerely,

Your Name

Don't worry about offending anyone. In fact, the school just might contact you and want to know what's better about the program in which you've decided to enroll. That happened to me, and I understood that the dean and faculty at the other schools really were interested in knowing what it was about their program that made it a less attractive option for me. They may want to consider hiring new faculty or otherwise improving their program.

(OR)

They would definitely appreciate knowing that you do not plan to attend, because they still have time to make another offer if they choose. At each program you won't be attending, you should email the faculty member with whom you had the most contact (or the central office, if that's how things were organized) with a short note, such as

"Dear [x], I really enjoyed meeting with you, and I was very excited to receive the offer of admission to [x]. However, I have decided that a different program is the best fit for me, and therefore, I am declining my offer of admission to your program. Thanks again for your consideration."

Obviously, rephrase to make it appropriate to your situation (reflecting your actual contact with them), but explaining your choice as a matter of better "fit" with another program, is a very socially acceptable phrase in academic circles.

I had to do this. As I recall, I wrote something like this:

Dear Whoever:

Thank you very much for your letter offering me admission to the PhD program in Whatever at Fabulous University. While I am grateful that the committee responded favorably to my application, I have received other offers. After much thought, I have decided that a program at another school will best address my interests.

Again, I very much appreciate your consideration and your offer. The decision was difficult, and I want to assure you that I do think very highly of the program at Fabulous University.

Sincerely,

Your Name

Don't worry about offending anyone. In fact, the school just might contact you and want to know what's better about the program in which you've decided to enroll. That happened to me, and I understood that the dean and faculty at the other schools really were interested in knowing what it was about their program that made it a less attractive option for me. They may want to consider hiring new faculty or otherwise improving their program.

They would definitely appreciate knowing that you do not plan to attend, because they still have time to make another offer if they choose. At each program you won't be attending, you should email the faculty member with whom you had the most contact (or the central office, if that's how things were organized) with a short note, such as

"Dear [x], I really enjoyed meeting with you, and I was very excited to receive the offer of admission to [x]. However, I have decided that a different program is the best fit for me, and therefore, I am declining my offer of admission to your program. Thanks again for your consideration."

Obviously, rephrase to make it appropriate to your situation (reflecting your actual contact with them), but explaining your choice as a matter of better "fit" with another program, is a very socially acceptable phrase in academic circles.

Include the last sentence and say that the school you've chosen is a better fit.