An FSEOG is for undergraduates with exceptional financial need -that is, students with the lowest Estimated Family Contributions — and gives priority to students who receive Federal Pell Grants. An FSEOG does not have to be paid back.
The U.S. Department of Education guarantees that each participating school will receive enough money to pay the Federal Pell Grants of its eligible students. There's no guarantee every eligible student will be able to receive an FSEOG; students at each school will be awarded these funds based on availability at that school.
You can receive between $100 and $4,000 a year, depending on when you apply, your need, the funding level of the school you're attending, and the policies of the financial aid office where you attend school.