An FSEOG (Federal Supplemental Educational Opportunity Grant) is for undergraduates with exceptional financial need, that is student's with the lowest "Expected Family Contribution" (EFC) level. For those of you not familiar with EFC, this is the US federal government's appraisal of how much money your family should be able to contribute towards your education. You can obtain your EFC by submiting a FAFSA form. About 6 weeks after you submit this form (or 2-3 weeks if you submit it by the web) you will get a "Student Aid Report" (SAR) that will show you your EFC.
Not all students that are eligible for financial aid will receive an FSEOG. This is because the US federal government provides funds to each school, each of which must then make the final allocation. Amounts that are awarded are typically between $1000 and $4000 a year. The amount you may receive depends on the financial aid policies of the school, when you apply, your degree of need, and the funding level of the school.
Source: The Student Guide, published by the US Department of Education. Note that federal financial aid programs are subject to change. Check with the school(s) of your choice to get the very latest information.