It depends on the type of federal student loan.
Federal Direct (subsidized or unsubsidized) loans are made out to the student, and in the student’s name. Therefore, you as the student are responsible for repaying them. If your parents co-sign your loan, however, they will be held responsible if you default (do not pay). For federal student loans, your parents are not required to co-sign (even if you are under 18), so this would be optional. (Private student loan companies, however, usually do require a co-signer).
Undergraduates may also be offered Federal Parent PLUS loans, which are given to your parents. Therefore, your parents are responsible for paying this back.