There are three main types of non-citizens that are attending college in the US right now. These are outlined below with a few details about their eligibility for federal/state and institutional aid.
These are students born outside the US who have traveled to the United States to attend college. They are required to have an F-1 VISA status and a valid I-20 issued by the institution.
These students are not eligible for federal or state aid, but would be eligible for institutional scholarships or some outside scholarships.
Generally these students have a legal residency status that allows them to be in the US legally without being a citizen. Usually these students will have one of the following statuses: Permeant Resident or Conditional Resident; and will have documentation of their lawful presence in the US.
These students may be eligible for federal or state aid, as well as institutional scholarships or some outside scholarships.
These students were born abroad and do not have a legal residency status in the US.
These students are not eligible for federal aid. Depending on the state, they may be eligible for state grants. The policy will vary from institution to institution, but many colleges offer merit, athletic and participation awards to non-citizen/undocumented students.
The federal student aid office has put together a comprehensive list of commonly asked questions regarding financial aid for undocumented students and/or parents.