Government grants are provided by either the state or federal government. The most well-known of these is the Federal Pell Grant, which is awarded by the federal government solely on the basis of need. Students can apply for the Pell Grant by filling out the Free Application for Federal Student Aid (FAFSA) online at www.fafsa.ed.gov. This will determine eligibility for various sources of federal assistance, including the Pell Grant.
The government also provides merit-based grants or grants designed for specific types of research. These can be found at www.ed.gov/fund/grant/apply/grantapps/index.html. The government lists the various types of discretionary grants available. This is particularly useful if you are applying for assistance with graduate work. There are a number of federal grants designed to fund students intending to research areas of importance to the government, such as disability research and educational improvement.
Privately funded grants are available from a variety of sources. A quick web search can point you to most of them, but, in order to avoid sifting through site after site, you would be well-advised to register for one of the various websites designed to catalogue available scholarships and grants. A useful site is FastWeb.com. Their service is free to use, and, once you complete a profile that details your background and interests, they will match you with various scholarships that you qualify for, making it significantly easier to sort through the quagmire of funding available.
Another useful webpage for finding student aid is CollegeBoard.com. The same people that created and currently administer the SAT also provide a comprehensive scholarship listing on their website. It is highly searchable and can be personalized in the same way FastWeb can, though unlike FastWeb, it is not required that you register before using the service. CollegeBoard also provides a number of other services to help you plan for your secondary education including SAT preparation, help on application essays and advice on how to prepare yourself throughout high school.
Lastly, most colleges and universities have some aid available to students. These can generally be found on an individual school's website along with other options for financial aid. Some advantages of applying to these scholarships is that the pool of applicants is restricted to the school and not the entire nation. Applying for these grants can also provide the opportunity to meet with administrators and do some networking before arriving on the scene.