The Free Application for Federal Student Aid is the form the U.S. Government requires students fill out to get federally sponsored financial aid. This aid can come in the form of student loans, such as Stafford, Perkins or PLUS loans, which must be paid back once you graduate at a relatively low interest rate; or grants, such as the Pell Grant, which do not need to be paid back. In both these cases, all aid can be used to cover housing costs.
Similar to grants, scholarships offer you money for your education that you do not need to pay back, but come from a private or even corporate donor, and must be applied for independently. While some scholarships have no strings attached, others will require you to only use them only for educational purposes, such as tuition fees or books, that don't include housing. In many cases, this money is given out incrementally, and if you break the terms, you may lose the remainder of the scholarship.
In the case of federally sponsored financial aid, half the amount of a loan or grant is given to your college or university at the beginning of each semester. There, it is directly applied to your tuition and any other costs you have with your school, which if you are staying on campus will include housing. However, if you are staying off-campus, you will receive the remainder of the aid, which you can then use at your discretion. But because of the time it takes to process, you may not be able to get this money until classes have already begun, so you may need to plan ahead for making your first month's rent.
While financial aid programs are usually enough to cover a good deal of the cost of housing and other related expenses, in many cases they will to be enough to cover all of them. If they are available, consider applying for a work-study program, which will require the school to give you a job. Even if you do not have a work-study program, it is not difficult for most students to find an on-campus job if they want one. These jobs are usually only minimum wage, but just 10 or 15 hours a week can go a long way toward covering your remaining housing costs.