The majority of students applying and receiving college grants are low income students, also called need-based grants. Need based college grants are to give everyone a fair opportunity to get an education. Some common need-based grants are the Pell Grant, a Scholarship for Disadvantaged Students, and several others. The first place you should apply is called the FAFSA.
Grants for doctoral candidates and graduate students are highly competitive, and focus on the financial needs of students engaged in research to complete their high level degrees. These grants are often referred to as fellowships, and are typically sponsored by colleges and universities as a way of bringing the best and brightest graduate students to their campuses. Unlike the more traditional undergraduate grants, these programs place a great amount of weight on academic achievement. Financial need is a secondary consideration. Grants for graduate and doctoral students are typically high dollar awards, and will include funds for research related travel and stipends for living expenses.