Federal government aid is provided by the federal (national) government. It is allocated based on the FAFSA form.
State aid (or state grants) is provided by your state’s government. It may be allocated based on the FAFSA or based on a separate state aid application. Each state has their own application process, but most states have one “main” application and then several separate applications for less common grants with narrow eligibility requirements (like ones specifically for foster children, or for children of fallen police officers/firefighters, etc.). Most state aid is reserved for students staying in-state for college.