I think this is how it works:
1) Complete the FAFSA and receive your Student Aid Report with your EFC.
2) Calculate: School's annual budget - EFC = eligibility for aid.
So, with an annual budget of $40,000 and an EFC of $10,000
$40,000 - $10,000 = $30,000
You are eligible for $30,000 in financial aid.
The first place to go for aid is always the government - ie. Stafford loans. There are two types - subsidized [the gov't pays your interest while you are in school] and unsubsidized [you pay your interest while in school - you can get this deferred until you graduate, though]. The max any student can get in Stafford loans is $18,500 ($8,500 is subsidized and $10,000 is unsubsidized).
3) Eligibility for aid - government aid (Stafford loans) = private loan eligibility
So, $30,000 - $18,500 = $11,500
You are eligible for $11,500 in private loans.
4) Apply for Stafford loans.
5) Apply for private loans.
The main confusing part for me was that my FAFSA and school financial aid applications were not actually applying for aid. Those applications are simply to determine eligibility for aid. Once that is determined, you still have to apply for the aid for which you are eligible. At some schools, after they have received your FAFSA info. and financial aid app., they will send you an award letter indicating eligibility and also including forms to begin applying for the loans for which you are eligible. (Often with detailed instructions). At other schools, the financial aid apps have a spot where you can check off and the school will initiate the loans (Stafford or private) for you upon receipt of your FAFSA and financial aid app (if you are eligible).