In short, the EFC is the amount of money the federal government expects you to put forth yearly for your education. This number is derived from a complicated formula that can be found at any school or online at the student aid on the web site.
The formula for the 2004-2005 school year has been published here.
The EFC is the number that your school receives from the federal government to indicate your financial need. The EFC is derived from your answers on the FAFSA form. When you are finished with your FAFSA, the EFC will be displayed on the confirmation page.
The school that you plan to attend will subtract your EFC from the cost of attendance, and the reminder will be your "financial need."
The school will then base your financial aid package on your financial need.
For example, if you plan on attending The University of Arizona, you may fill out your FAFSA application and receive and EFC of $3,500. If the cost of attendance at University of Arizona is $8,000 per year, then your financial need is $4,500.