Although you should apply to more than one university, you ought to do enough research that you have a firm idea of the three programs which would suit your needs.
1. Study the department websites.
2. Register for a GRE test offered at least six weeks prior to the application deadline.
3. Compose your statement of purpose. Ask advice from Department faculty.
4. Line up letters of recommendation. Three is usually the norm.
5. Complete and send all admission and financial aid applications ahead of the deadlines.
Most applications will request that the Registrar send an official transcript; you might also want to obtain a couple for yourself so that you can include them in applications. If you have taken courses at another institution, you will want to obtain an official transcript from there as well. FYI, you will need at least a GPA of 3.0 for acceptance to any reputable graduate program.
Most general graduate schools require the GRE. Be sure to make yourself familiar with the format of the GREs, and take as many sample tests as possible.
Statement of Purpose
This is possibly the most important aspect of the application. You should make sure that it reflects who you are and what your capabilities are. There is no room for careless mistakes; it must be read and proof-read by as many people as possible.
It is sometimes necessary to tailor each statement to the institution for which it is intended. Remember that it will be read by a number of individuals, not just the professor with whom you wish to study. You will want, therefore, to appear focused, but not narrow. On the other hand, you will want to demonstrate that you are prepared to engage in advanced research. This is a professional document; it must demonstrate a mature understanding of the field.
The opening sentence is especially important and must be eye-catching. In the first paragraph you should endeavor to convey your informed enthusiasm; in the second, you should communicate as much data as possible about yourself and your long-term goals; finally, you will want to express why it is that program X will enable you to make the most of your talents and achieve your goals.
Leave out folksy tales, experiences from youth or high school, and any negative aspects of your academic record: be positive.
The process or writing a personal statement takes time. You will want to show it to a professor, and you must expect to rewrite it many times before it is perfect. Do not leave it until the last minute. Begin brainstorming as soon as possible.
Letters of Recommendation
You will normally be asked to include with your application, or have sent separately, three or four letters of recommendation. These are very important. Try to have professors with whom you have taken a seminar, or with whom you have been on an FSP write for you. The more familiar they are with you, the more effective their letter will be. Specific examples are necessary; you might have to remind the professor of a paper you wrote, or a presentation you delivered in class. Also, if there is a negative aspect of your performance at Dartmouth, it is best addressed in a letter of recommendation (rather than in a personal statement); discuss this with your professors. Also, it will help if those reading the letters know, or know of the professor who wrote them. Try to choose professors who are well-known in their fields.
When requesting that a professor write a letter of recommendation, the professor is both supporting you, and your application. Therefore, be sure to go, in person, and discuss your plans with professors as early in the process as possible. Also, show up prepared. Demonstrate that you have already put some thought into the matter (i.e., have some ideas about what you would like to study)
When you finally decide to which programs you will apply, supply your letter writers with:
a. a curriculum vitae/resumé
b. a transcript or list of courses you have taken at Dartmouth
c. explicit information about the date the letter is due, and how the letter is to be sent to the program.
d. your statement of purpose