Basic Tips for the Application for Graduate School

General Tips:

Turn it in on time (give those professors plenty of time - and reminders - to write those letters). Take the requisite tests.
They like to see advanced courses and language proficiency (reading level).
Always apply for financial aid.
List every little award and honor.
Spend some time and thought on the essay.
Take advanced courses

Be as specific as possible about your goals and how THIS graduate program will help you achieve them.
Find out as much as you can about the school, the faculty, their research. Is there one particular person or area that catches your fancy?
Tailor each application to the specific school and program.
Talk yourself up - this isn’t the time to be shy! Did you do a research project or clinic that inspired you? Did you have a good teaching/mentoring experience?
If there is anything on the “lists” included in the rest of the application you’d like to explain, you should do it here: job experience, strange classes, anomalous grades, other activities not mentioned.

Inform your letter writers, too: if possible, give them a copy of everything - grades, awards, essay, the works.

Learn as much as you can about the different possibilities in the mathematical sciences. Read journals, books, talk to profs, go to colloquia.